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I 


•xSm*^i?c?^n 


A  TREATISE  ON  THE 

DIFFERENTIAL  GEOMETRY 

OF  PURVES  AND  SURFACES 


BY 
LUTHER  PFAHLER  EISENHART 

PROFESSOR   OF   MATHEMATICS   IN    PRINCETON   UNIVERSITY 


GINN  AND  COMPANY 

BOSTON  •  NEW  YORK  •  CHICAGO  •  LONDON 


CvA 


COPYRIGHT,  1909,  BY 
LUTHER  PFAHLER  EJSENHART 


ALL  RIGHTS   RESERVED 
89-8 


gftc   SUftengum 

GINN   AND   COMPANY  •  PKO- 
PRILTORS  •  BOSTON  •  U.S.A. 


£6 

A-.ATH.. 

STAT. 

LIBRARY 


PEEFACE 


This  book  is  a  development  from  courses  which  I  have  given  in 
Princeton  for  a  number  of  years.  During  this  time  I  have  come  to 
feel  that  more  would  be  accomplished  by  my  students  if  they  had  an 
introductory  treatise  written  in  English  and  otherwise  adapted  to  the 
use  of  men  beginning  their  graduate  work. 

Chapter  I  is  devoted  to  the  theory  of  twisted  curves,  the  method 
in  general  being  that  which  is  usually  followed  in  discussions  of  this 
subject.  But  in  addition  I  have  introduced  the  idea  of  moving  axes, 
and  have  derived  the  formulas  pertaining  thereto  from  the  previously 
obtained  Frenet-Serret  formulas.  In  this  way  the  student  is  made 
familiar  with  a  method  which  is  similar  to  that  used  by  Darboux  in 
the  first  volume  of  his  Lemons,  and  to  that  of  Cesaro  in  his  Geometria 
Intrinseca.  This  method  is  not  only  of  great  advantage  in  the  treat 
ment  of  certain  topics  and  in  the  solution  of  problems,  but  it  is  valu 
able  in  developing  geometrical  thinking. 

The  remainder  of  the  book  may  be  divided  into  three  parts.  The 
first,  consisting  of  Chapters  II-YI,  deals  with  the  geometry  of  a  sur 
face  in  the  neighborhood  of  a  point  and  the  developments  therefrom, 
such  as  curves  and  systems  of  curves  defined  by  differential  equa 
tions.  To  a  large  extent  the  method  is  that  of  Gauss,  by  which  the 
properties  of  a  surface  are  derived  from  the  discussion  of  two  quad 
ratic  differential  forms.  However,  little  or  no  space  is  given  to  the 
algebraic  treatment  of  differential  forms  and  their  invariants.  In 
addition,  the  method  of  moving  axes,  as  defined  in  the  first  chapter, 
has  been  extended  so  as  to  be  applicable  to  an  investigation  of  the 
properties  of  surfaces  and  groups  of  surfaces.  The  extent  of  the 
theory  concerning  ordinary  points  is  so  great  that  no  attempt  has 
been  made  to  consider  the  exceptional  problems.  For  a  discussion 
of  such  questions  as  the  existence  of  integrals  of  differential  equa 
tions  and  boundary  conditions  the  reader  must  consult  the  treatises 
which  deal  particularly  with  these  subjects. 

In  Chapters  VII  and  VIII  the  theory  previously  developed  is 
applied  to  several  groups  of  surfaces,  such  as  the  quadrics,  ruled 
surfaces,  minimal  surfaces,  surfaces  of  constant  total  curvature,  and 
surfaces  with  plane  and  spherical  lines  of  curvature. 

iii 


iv  PEEFACE 

The  idea  of  applicability  of  surfaces  is  introduced  in  Chapter  III 
as  a  particular  case  of  conformal  representation,  and  throughout  the 
book  attention  is  called  to  examples  of  applicable  surfaces.  However, 
the  general  problems  concerned  with  the  applicability  of  surfaces  are 
discussed  in  Chapters  IX  and  X,  the  latter  of  which  deals  entirely 
with  the  recent  method  of  Weingarten  and  its  developments.  The 
remaining  four  chapters  are  devoted  to  a  discussion  of  infinitesimal 
deformation  of  surfaces,  congruences  of  straight  lines  and  of  circles, 
and  triply  orthogonal  systems  of  surfaces. 

It  will  be  noticed  that  the  book  contains  many  examples,  and  the 
student  will  find  that  wrhereas  certain  of  them  are  merely  direct 
applications  of  the  formulas,  others  constitute  extensions  of  the 
theory  which  might  properly  be  included  as  portions  of  a  more  ex 
tensive  treatise.  At  first  I  felt  constrained  to  give  such  references  as 
would  enable  the  reader  to  consult  the  journals  and  treatises  from 
which  some  of  these  problems  were  taken,  but  finally  it  seemed  best 
to  furnish  no  such  key,  only  to  remark  that  the  Encyklopadie  der 
mathematisclicn  Wissenschaften  may  be  of  assistance.  And  the  same 
may  be  said  about  references  to  the  sources  of  the  subject-matter  of 
the  book.  Many  important  citations  have  been  made,  but  there  has 
not  been  an  attempt  to  give  every  reference.  However,  I  desire  to 
acknowledge  my  indebtedness  to  the  treatises  of  Darboux,  Bianchi, 
and  Scheffers.  But  the  difficulty  is  that  for  many  years  I  have  con 
sulted  these  authors  so  freely  that  now  it  is  impossible  for  me  to  say, 
except  in  certain  cases,  what  specific  debts  I  owe  to  each. 

In  its  present  form,  the  material  of  the  first  eight  chapters  has 
been  given  to  beginning  classes  in  each  of  the  last  two  years;  and 
the  remainder  of  the  book,  with  certain  enlargements,  has  constituted 
an  advanced  course  which  has  been  followed  several  times.  It  is  im 
possible  for  me  to  give  suitable  credit  for  the  suggestions  made  and 
the  assistance  rendered  by  my  students  during  these  years,  but  I  am 
conscious  of  helpful  suggestions  made  by  my  colleagues,  Professors 
Veblen,  Maclnnes,  and  Swift,  and  by  my  former  colleague,  Professor 
Bliss  of  Chicago.  I  wish  also  to  thank  Mr.  A.  K.  Krause  for  making 
the  drawings  for  the  figures. 

It  remains  for  me  to  express  my  appreciation  of  the  courtesy 
shown  by  Ginn  and  Company,  and  of  the  assistance  given  by  them 
during  the  printing  of  this  book. 

LUTHER  PFAHLER  E1SENHART 


CONTENTS 


CHAPTEE  I 

CURVES  IN  SPACE 
SECTION  PAGE 

1.  PARAMETRIC  EQUATIONS  OF  A  CURVE 1 

2.  OTHER  FORMS  OF  THE  EQUATIONS  OF  A  CURVE 3 

3.  LINEAR  ELEMENT 4 

4.  TANGENT  TO  A  CURVE 6 

5.  ORDER  OF  CONTACT.    NORMAL  PLANK 8 

6.  CURVATURE.    RADIUS  OF  FIRST  CURVATURE 9 

7.  OSCULATING  PLANE 10 

8.  PRINCIPAL  NORMAL  AND  BINORMAL 12 

9.  OSCULATING  CIRCLE.    CENTER  OF  FIRST  CURVATURE     ....  14 

10.  TORSION.    FRENET-SERRET  FORMULAS 16 

11.  FORM  OF  CURVE  IN  THE  NEIGHBORHOOD  OF  A  POINT.    THE  SIGN 

OF  TORSION 18 

12.  CYLINDRICAL  HELICES 20 

13.  INTRINSIC  EQUATIONS.    FUNDAMENTAL  THEOREM 22 

14.  RICCATI  EQUATIONS 25 

15.  THE  DETERMINATION  OF  THE  COORDINATES  OF  A  CURVE  DEFINED 

BY  ITS  INTRINSIC  EQUATIONS 27 

16.  MOVING  TRIHEDRAL 30 

17.  ILLUSTRATIVE  EXAMPLES .33 

18.  OSCULATING  SPHERE 37 

19.  BERTRAND  CURVES 39 

20.  TANGENT  SURFACE  OF  A  CURVE 41 

21.  INVOLUTES  AND  EVOLUTES  OF  A  CURVE 43 

22.  MINIMAL  CURVES                                                                                  .  47 


CHAPTER  II 
CURVILINEAR  COORDINATES  ON  A  SURFACE.    ENVELOPES 

23.  PARAMETRIC  EQUATIONS  OF  A  SURFACE 52 

24.  PARAMETRIC  CURVES 54 

25.  TANGENT  PLANE 56 

26.  ONE-PARAMETER  FAMILIES  OF  SURFACES.    ENVELOPES    ....  59 

v 


vi  CONTENTS 

SECTION  PAGE 

27.  DEVELOPABLE  SURFACES.    RECTIFYING  DEVELOPABLE    ....       61 

28.  APPLICATIONS  OF  THE  MOVING  TRIHEDRAL 04 

29.  ENVELOPE  OF  SPHERES.    CANAL  SURFACES  66 


CHAPTER  III 

LINEAR  ELEMENT  OF  A  SURFACE.    DIFFERENTIAL  PARAME 
TERS.    CONFORMAL  REPRESENTATION 

30.  LINEAR  ELEMENT 70 

31.  ISOTROPIC  DEVELOPABLE 72 

32.  TRANSFORMATION  OF  COORDINATES 72 

33.  ANGLES  BETWEEN  CURVES.    THE  ELEMENT  OF  AREA    ....  74 

34.  FAMILIES  OF  CURVES 78 

35.  MINIMAL  CURVES  ON  A  SURFACE 81 

36.  VARIATION  OF  A  FUNCTION 82 

37.  DIFFERENTIAL  PARAMETERS  OF  THE  FIRST  ORDER 84 

38.  DIFFERENTIAL  PARAMETERS  OF  THE  SECOND  ORDER    ....  88 

39.  SYMMETRIC  COORDINATES 91 

40.  ISOTHERMIC  AND  ISOMETRIC  PARAMETERS 93 

41.  ISOTHERMIC  ORTHOGONAL  SYSTEMS 95 

42.  CONFORMAL  REPRESENTATION 98 

43.  ISOMETRIC  REPRESENTATION.    APPLICABLE  SURFACES    ....  100 

44.  CONFORMAL  REPRESENTATION  OF  A  SURFACE  UPON  ITSELF  .     .  101 

45.  CONFORMAL  REPRESENTATION  OF  THE  PLANE 104 

46.  SURFACES  OF  REVOLUTION ' 107 

47.  CONFORMAL  REPRESENTATIONS  OF  THE  SPHERE    .  109 


CHAPTER  IV 

GEOMETRY  OF  A  SURFACE  IN  THE  NEIGHBORHOOD 
OF  A  POINT 

48.  FUNDAMENTAL  COEFFICIENTS  OF  THE  SECOND  ORDER  ....  114 

49.  RADIUS  OF  NORMAL  CURVATURE 117 

50.  PRINCIPAL  RADII  OF  NORMAL  CURVATURE 118 

51.  LINES  OF  CURVATURE.    EQUATIONS  OF  RODRIGUES 121 

52.  TOTAL  AND  MEAN  CURVATURE 123 

53.  EQUATION  OF  EULER.    DUPIN  INDICATRIX 124 

54.  CONJUGATE  DIRECTIONS  AT  A  POINT.    CONJUGATE  SYSTEMS      .  126 

55.  ASYMPTOTIC  LINES.    CHARACTERISTIC  LINES     .......  128 

56.  CORRESPONDING  SYSTEMS  ON  Two  SURFACES 130 

57.  GEODESIC  CURVATURE.    GEODESICS ...  131 

58.  FUNDAMENTAL  FORMULAS  .  133 


CONTENTS  vii 

SECTION  PAGE 

59.    GEODESIC  TORSION 137 

GO.    SPHERICAL  REPRESENTATION 141 

61.  RELATIONS   BETWEEN  A   SURFACE   AND   ITS   SPHERICAL    REPRE 

SENTATION  143 

62.  HELICOIDS  146 


CHAPTEE  V 

FUNDAMENTAL  EQUATIONS.    THE  MOVING  TRIHEDRAL 

63.  ClIRISTOFFEL  SYMBOLS 152 

64.  THE  EQUATIONS  OF  GAUSS  AND  OF  CODAZZI 153 

65.  FUNDAMENTAL  THEOREM 157 

66.  FUNDAMENTAL  EQUATIONS  IN  ANOTHER  FORM 160 

67.  TANGENTIAL  COORDINATES.    MEAN  EVOLUTE 162 

68.  THE  MOVING  TRIHEDRAL 166 

69.  FUNDAMENTAL  EQUATIONS  OF  CONDITION 168 

70.  LINEAR  ELEMENT.    LINES  OF  CURVATURE 171 

71.  CONJUGATE    DIRECTIONS  AND  ASYMPTOTIC   DIRECTONS.    SPHER 

ICAL  REPRESENTATION 172 

72.  FUNDAMENTAL  RELATIONS  AND  FORMULAS 174 

73.  PARALLEL  SURFACES 177 

74.  SURFACES  OF  CENTER 179 

75.  FUNDAMENTAL  QUANTITIES  FOR  SURFACES  OF  CENTER     .     .     .  181 

76.  SURFACES  COMPLEMENTARY  TO  A  GIVEN  SURFACE 184 

CHAPTER  VI 

SYSTEMS  OF  CURVES.    GEODESICS 

77.  ASYMPTOTIC  LINES 189 

78.  SPHERICAL  REPRESENTATION  OF  ASYMPTOTIC  LINES      ....  191 

79.  FORMULAS  OF  LELIEUVRE.    TANGENTIAL  EQUATIONS    ....  193 

80.  CONJUGATE  SYSTEMS  OF  PARAMETRIC  LINES.    INVERSIONS    .     .  195 

81.  SURFACES  OF  TRANSLATION 197 

82.  ISOTHERMAL-CONJUGATE  SYSTEMS 198 

83.  SPHERICAL  REPRESENTATION  OF  CONJUGATE  SYSTEMS       .     .     .  200 

84.  TANGENTIAL  COORDINATES.    PROJECTIVE  TRANSFORMATIONS      .  201 

85.  EQUATIONS  OF  GEODESIC  LINES 204 

86.  GEODESIC  PARALLELS.    GEODESIC  PARAMETERS 206 

87.  GEODESIC  POLAR  COORDINATES 207 

88.  AREA  OF  A  GEODESIC  TRIANGLE 209 

89.  LINES  OF  SHORTEST  LENGTH.    GEODESIC  CURVATURE  ....  212 

90.  GEODESIC  ELLIPSES  AND  HYPERBOLAS                                          .     .  213 


viii  CONTENTS 

SECTION  PAGE 

91.  SURFACES  OF  LIOUVILLE 214 

92.  INTEGRATION  OF  THE  EQUATION  OF  GEODESIC  LINES       .     .     .     215 

93.  GEODESICS  ON  SURFACES  OF  LIOUVILLE 218 

94.  LINES  OF  SHORTEST  LENGTH.    ENVELOPE  OF  GEODESICS  220 


CHAPTEK  VII 

QUADRICS.    RULED  SURFACES.    MINIMAL   SURFACES 

95.  CONFOCAL  QUADRICS.    ELLIPTIC  COORDINATES 226 

96.  FUNDAMENTAL  QUANTITIES  FOR  CENTRAL  QUADRICS       .     .     .  229 

97.  FUNDAMENTAL  QUANTITIES  FOR  THE  PARABOLOIDS     ....  230 

98.  LINES  OF  CURVATURE  AND  ASYMPTOTIC  LINES  ox  QUADRICS    .  232 

99.  GEODESICS  ON  QUADRICS 231 

100.  GEODESICS  THROUGH  THE  UMBILICAL  POINTS 236 

101.  ELLIPSOID  REFERRED  TO  A  POLAR  GEODESIC  SYSTEM       .     .     .  236 

102.  PROPERTIES  OF  QUADRICS 239 

103.  EQUATIONS  OF  A  RULED  SURFACE 241 

104.  LINE  OF  STRICTION.    DEVELOPABLE  SURFACES 242 

105.  CENTRAL  PLANE.    PARAMETER  OF  DISTRIBUTION 244 

106.  PARTICULAR  FORM  OF  THE  LINEAR  ELEMENT 247 

107.  ASYMPTOTIC  LINES.    ORTHOGONAL  PARAMETRIC  SYSTEMS    .     .  248 

108.  MINIMAL  SURFACES 250 

109.  LINES  OF  CURVATURE  AND  ASYMPTOTIC  LINES.    ADJOINT  MINI 

MAL  SURFACES 253 

110.  MINIMAL  CURVES  ON  A  MINIMAL  SURFACE 254 

111.  DOUBLE  MINIMAL  SURFACES 258 

112.  ALGEBRAIC  MINIMAL  SURFACES 260 

113.  ASSOCIATE  SURFACES 263 

114.  FORMULAS  OF  SCHWARZ 264 

.  r 

CHAPTER  VIII 

SURFACES  OF  CONSTANT  TOTAL  CURVATURE.  TF-SURFACES. 
SURFACES  WITH  PLANE  OR  SPHERICAL  LINES  OF  CUR 
VATURE 

115.  SPHERICAL  SURFACES  OF  REVOLUTION 270 

116.  PSEUDOSPHERICAL  SURFACES  OF  REVOLUTION 272 

117.  GEODESIC  PARAMETRIC  SYSTEMS.    APPLICABILITY 275 

118.  TRANSFORMATION  OF  HAZZIDAKIS 278 

119.  TRANSFORMATION  OF  BIANCHI 280 

120.  TRANSFORMATION  OF  BACKLUND 284 

121.  THEOREM  OF  PERMUTABILITY 286 


CONTENTS  ix 

SECTION  PAGE 

122.  TRANSFORMATION  OF  LIE •    .     .  289 

123.  JF-SURFACES.    FUNDAMENTAL  QUANTITIES 291 

124.  EVOLUTE  OF  A  TF-SuitFACE .  294 

125.  SURFACES  OF  CONSTANT  MEAN  CURVATURE    .......  296 

126.  RULED  IF-SuRFACES 299 

127.  SPHERICAL  REPRESENTATION  OF  SURFACES  WITH  PLANE  LINES 

OF  CURVATURE  IN  BOTH  SYSTEMS 300 

128.  SURFACES  WITH  PLANE  LINES  OF  CURVATURE  IN  BOTH  SYSTEMS  302 

129.  SURFACES  WITH  PLANE  LINES  OF  CURVATURE  IN  ONE  SYSTEM. 

SURFACES  OF  MONGE 305 

130.  MOLDING  SURFACES 307 

131.  SURFACES  OF  JOACHIMSTHAL 308 

132.  SURFACES  WITH  CIRCULAR  LINES  OF  CURVATURE 310 

133.  CYCLIDES  OF  DUPIN 312 

134.  SURFACES   WITH    SPHERICAL    LINES    OF    CURVATURE    IN  ONE 

SYSTEM                      314 


CHAPTER  IX 
DEFORMATION  OF   SURFACES 

135.  PROBLEM  OF  MINDING.    SURFACES  OF  CONSTANT  CURVATURE  .  321 

136.  SOLUTION  OF  THE  PROBLEM  OF  MINDING 323 

137.  DEFORMATION  OF  MINIMAL  SURFACES 327 

138.  SECOND  GENERAL  PROBLEM  OF  DEFORMATION 331 

139.  DEFORMATIONS   WHICH    CHANGE   A    CURVE   ON   THE   SURFACE 

INTO  A  GIVEN  CURVE  IN  SPACE 333 

140.  LINES  OF  CURVATURE  IN  CORRESPONDENCE 336 

141.  CONJUGATE  SYSTEMS  IN  CORRESPONDENCE 338 

142.  ASYMPTOTIC  LINES  IN  CORRESPONDENCE.    DEFORMATION  OF  A 

RULED  SURFACE 342 

143.  METHOD  OF  MINDING 344 

144.  PARTICULAR  DEFORMATIONS  OF  RULED  SURFACES 345 

CHAPTER  X 
DEFORMATION  OF  SURFACES.    THE  METHOD  OF  WEINGARTEN 

145.  REDUCED  FORM  OF  THE  LINEAR  ELEMENT 351 

146.  GENERAL  FORMULAS 353 

147.  THE  THEOREM  OF  WEINGARTEN 355 

148.  OTHER  FORMS  OF  THE  THEOREM  OF  WEINGARTEN      ....  357 

149.  SURFACES  APPLICABLE  TO  A  SURFACE  OF  REVOLUTION        .     .  362 


x  CONTENTS 

SECTION  PAGE 

150.  MINIMAL  LINES  ON  THE  SPHERE  PARAMETRIC 364 

151.  SURFACES  OF   GOURSAT.     SURFACES  APPLICABLE   TO  CERTAIN 

PARABOLOIDS 366 

CHAPTER   XI 

INFINITESIMAL  DEFORMATION  OF  SURFACES 

152.  GENERAL  PROBLEM 373 

153.  CHARACTERISTIC  FUNCTION ....  374 

154.  ASYMPTOTIC  LINES  PARAMETRIC 376 

155.  ASSOCIATE  SURFACES 378 

156.  PARTICULAR  PARAMETRIC  CURVES 379 

157.  RELATIONS  BETWEEN  THREE  SURFACES  S,  Sv  S0 382 

158.  SURFACES  RESULTING  FROM  AN  INFINITESIMAL  DEFORMATION  385 

159.  ISOTHERMIC  SURFACES       387 

CHAPTER  XII 
RECTILINEAR  CONGRUENCES 

160.  DEFINITION  OF  A  CONGRUENCE.     SPHERICAL   REPRESENTATION  392 

161.  NORMAL  CONGRUENCES.    RULED  SURFACES  OF  A  CONGRUENCE  393 

162.  LIMIT  POINTS.    PRINCIPAL  SURFACES 395 

163.  DEVELOPABLE  SURFACES  OF  A  CONGRUENCE.    FOCAL  SURFACES  398 

164.  ASSOCIATE  NORMAL  CONGRUENCES 401 

165.  DERIVED  CONGRUENCES 403 

166.  FUNDAMENTAL  EQUATIONS  OF  CONDITION  ........  406 

167.  SPHERICAL  REPRESENTATION  OF  PRINCIPAL  SURFACES  AND  OF 

DEVELOPABLES        ......  407 

168.  FUNDAMENTAL  QUANTITIES  FOR  THE  FOCAL  SURFACES  .     .     .  409 

169.  ISOTROPIC  CONGRUENCES 412 

170.  CONGRUENCES  OF  GUICIIARD 414 

171.  PSEUDOSPHERICAL    CONGRUENCES 415 

172.  IT-CONGRUENCES 417 

173.  CONGRUENCES  OF  RIBAUCOUR 420 

CHAPTER  XIII 
CYCLIC  SYSTEMS 

174.  GENERAL  EQUATIONS  OF  CYCLIC  SYSTEMS 426 

175.  CYCLIC  CONGRUENCES 431 

176.  SPHERICAL  REPRESENTATION  OF  CYCLIC  CONGRUENCES   .  432 


CONTENTS 


XI 


SECTION  PAGE 

177.  SURFACES  ORTHOGONAL  TO  A  CYCLIC  SYSTEM 436 

178.  NORMAL  CYCLIC  CONGRUENCES 437 

179.  CYCLIC  SYSTEMS  FOR  WHICH  THE   ENVELOPE  OF  THE  PLANES 

OF  THE  CIRCLES  is  A  CURVE 439 

180.  CYCLIC   SYSTEMS   FOR   WHICH   THE    PLANES   OF    THE    CIRCLES 

PASS  THROUGH  A  POINT 440 

CHAPTEE  XIV 

TRIPLY  ORTHOGONAL  SYSTEMS  OF  SURFACES 

181.  TRIPLE    SYSTEM    OF    SURFACES    ASSOCIATED    WITH    A    CYCLIC 

SYSTEM 446 

182.  GENERAL  EQUATIONS.    THEOREM  OF  DUPIN 447 

183.  EQUATIONS  OF  LAME 449 

184.  TRIPLE    SYSTEMS  CONTAINING   ONE   FAMILY  OF    SURFACES  OF 

REVOLUTION 451 

185.  TRIPLE  SYSTEMS  OF  BIANCHI  AND  OF  WEINGARTEN  ....  452 

186.  THEOREM  OF  RIBAUCOUR 457 

187.  THEOREMS  OF  DARBOUX 458 

188.  TRANSFORMATION  OF  COMBESCURE 461 

INDEX   .  467 


DIFFERENTIAL  GEOMETRY 


CHAPTER  I 

CURVES  IN  SPACE 

1.  Parametric  equations  of  a  curve.  Consider  space  referred  to 
fixed  rectangular  axes,  and  let  (x,  y,  z)  denote  as  usual  the  coordi 
nates  of  a  point  with  respect  to  these  axes.  In  the  plane  2  =  0 
draw  a  circle  of  radius  r  and  center  (a,  b).  The  coordinates  of  a 
point  P  on  the  circle  can  be  expressed  in  the  form 

(1)  x  —  a  -{-  r  cos  u,     y  =  b  H-  r  sin  u,     2  =  0, 

where  u  denotes  the  angle  which  the  radius  to  P  makes  with  the 
positive  o>axis.  As  u  varies  from  0°  to  360°,  the  point  P  describes 
the  circle.  The  quantities  a,  5,  r  determine  the  position  and  size 
of  the  circle,  whereas  u  determines  the  position  of  a  point  upon  it. 
In  this  sense  it  is  a  variable  or  parameter  for  the 
circle.  And  equations  (1)  are  called  parametric 
equations  of  the  circle. 

A  straight  line  in  space  is  determined  by  a 
point  on  it,  PQ(a,  6,  c),  and  its  direction-cosines 
a,  /3,  7.  The  latter  fix  also  the  sense  of  the  line. 
Let  P  be  another  point  on  the  line,  and  let  the 
distance  PQP  be  denoted  by  u,  which  is  positive 
or  negative.  •  The  rectangular  coordinates  of  P 
are  then  expressible  in  the  form 

(2)  x  =  a  +  ua,     y  =  b  +  u(B,    z  —  c  +  wy. 

To  each  value  of  u  there  corresponds  a  point 
on  the  line,  and  the  coordinates  of  any  point  on  the  line  are 
expressible  as  in  (2).  These  equations  are  consequently  parametric 
equations  of  the  straight  line. 

When,  as  in  fig.  1,  a  line  segment  PD,  of  constant  length  «,  per 
pendicular  to  a  line  OZ  at  D,  revolves  uniformly  about  OZ  as  axis, 


FIG.  1 


2  CURVES  IN  SPACE 

and  at  the  same  time  D  moves  along  it  with  uniform  velocity,  the 
locus  of  P  is  called  a  circular  helix.  If  the  line  OZ  be  taken  for  the 
2-axis,  the  initial  position  of  PD  for  the  positive  a>axis,  and  the  angle 
between  the  latter  and  a  subsequent  position  of  PD  be  denoted  by 
u,  the  equations  of  the  helix  can  be  written  in  the  parametric  form 

(3)  x  =  a  cos  u,     y  —  a  sin  u,     z  =  bu, 

where  the  constant  b  is  determined  by  the  velocity  of  rotation  of 
PD  and  of  translation  of  D.  Thus,  as  the  line  PD  describes  a 
radian,  D  moves  the  distance  b  along  OZ. 

In  all  of  the  above  equations  u  is  the  variable  or  parameter. 
Hence,  with  reference  to  the  locus  under  consideration,  the  coordi 
nates  are  functions  of  u  alone.  We  indicate  this  by  writing  these 
equations 

The  functions  /x,  /2,  /„  have  definite  forms  when  the  locus  is  a 
circle,  straight  line  or  circular  helix.  But  we  proceed  to  the  gen 
eral  case  and  consider  equations  (4),  when  /r  /2,  /„  are  any  func 
tions  whatever,  analytic  for  all  values  of  u,  or  at  least  for  a  certain 
domain.*  The  locus  of  the  point  whose  coordinates  are  given  by  (4), 
as  u  takes  all  values  in  the  domain  considered,  is  a  curve.  Equa 
tions  (4)  are  said  to  be  the  equations  of  the  curve  in  the  parametric 
form.  When  all  the  points  of  the  curve  do  not  lie  in  the  same  plane 
it  is  called  a  space  curve  or  a  twisted  curve ;  otherwise,  a  plane  curve. 
It  is  evident  that  a  necessary  and  sufficient  condition  that  a 
curve,  defined  by  equations  (4),  be  plane,  is  that  there  exist  a 
linear  relation  between  the  functions,  such  as 

(5)  of i  +  5f2  +  c/3  +  d  =  0, 

where  a,  b,  c,  d  denote  constants  not  all  equal  to  zero.    This  con 
dition  is  satisfied  by  equations  (1)  and  (2),  but  not  by  (3). 
If  u  in  (4)  be  replaced  by  any  function  of  v,  say 

(6)  *fc*^(*0, 

equations  (4)  assume  a  new  form, 

*  E.g.  in  case  u  is  supposed  to  be  real,  it  lies  on  a  segment  between  two  fixed  values; 
when  it  is  complex,  it  lies  within  a  closed  region  in  the  plane  of  the  complex  variable. 


EQUATIONS  OF  A  CURVE  3 

It  is  evident  that  the  values  of  x,  y,  z,  given  by  (7)  for  a  value 
of  i',  are  equal  to  those  given  by  (4)  for  the  corresponding  value 
of  u  obtained  from  (6).  Consequently  equations  (4)  and  (7)  define 
the  same  curve,  u  and  v  being  the  respective  parameters.  Since 
there  is  no  restriction  upon  the  function  </>,  except  that  it  be  ana 
lytic,  it  follows  that  a  curve  can  be  given  parametric  representation 
in  an  infinity  of  ways. 

2.  Other  forms  of  the  equations  of  a  curve.  If  the  first  of  equa 
tions  (4)  be  solved  for  w,  giving  u  —  $(#),  then,  in  terms  of  x  as 
parameter,  equations  (7)  are 

(8)  x  =  x,     y  =  F2(x),     z  =  F8(x). 

In  this  form  the  curve  is  really  defined  by  the  last  two  equations, 
or,  if  it  be  a  plane  curve  in  the  o?y-plane,  its  equation  is  in  the 
customary  form 

(9)  y  =/(*)• 

The  points  in  space  whose  coordinates  satisfy  the  equation 
y  =  Fz(x)  lie  on  the  cylinder  whose  elements  are  parallel  to  the 
2-axis  and  whose  cross  section  by  the  xy-pl&ne  is  the  curve  y  =  F2(x). 
In  like  manner,  the  equation  z  =  F3(x)  defines  a  cylinder  whose 
elements  are  parallel  to  the  #-axis.  Hence  the  curve  with  the 
equations  (8)  is  the  locus  of  points  common  to  two  cylinders 
with  perpendicular  axes.  Conversely,  if  lines  are  drawn  through 
the  points  of  a  space  curve  normal  to  two  planes  perpendicular 
to  one  another,  we  obtain  two  such  cylinders  whose  intersection 
is  the  given  curve.  Hence  equations  (8)  furnish  a  perfectly  gen 
eral  definition  of  a  space  curve. 

In  general,  the  parameter  u  can  be  eliminated  from  equations  (4) 
in  such  a  way  that  there  result  two  equations,  each  of  which  in 
volves  all  three  rectangular  coordinates.  Thus, 

(10)  Qfa  y,  z)  =  0,          <S>a(a;,  y,  z)  =  0. 

Moreover,  if  two  equations  of  this  kind  be  solved  for  y  and  z  as 
functions  of  x,  we  get  equations  of  the  form  (8),  and,  in  turn,  of 
the  form  (4),  by  replacing  x  by  an  arbitrary  function  of  u.  Hence 
equations  (10)  also  are  the  general  equations  of  a  curve.  It  will 
be  seen  later  that  each  of  these  equations  defines  a  surface. 


4  CURVES  IN  SPACE 

It  should  be  remarked,  however,  that  when  a  curve  is  defined 
as  the  intersection  of  two  cylinders  (8),  or  of  two  surfaces  (10),  it 
may  happen  that  these  curves  of  intersection  consist  of  several 
parts,  so  that  the  new  equations  define  more  than  the  original  ones. 

For  example,  the  curve  defined  by  the  parametric  equations 
(i)  x  =  w,         y  =  w2,        z  =  w3, 

is  a  twisted  cubic,  for  every  plane  meets  the  curve  in  three  points.    Thus,  the  plane 

ax  +  by  -f  cz  +  d  =  0 

meets  the  curve  in  the  three  points  whose  parametric  values  are  the  roots  of  the 
e(luation  CM* +  &n"  + an +  d  =  0. 

This  cubic  lies  upon  the  three  cylinders 

y  =  x2,         z  =  x3,        y3  =  z2. 

The  intersection  of  the  first  and  second  cylinders  is  a  curve  of  the  sixth  degree, 
of  the  first  and  third  it  is  of  the  sixth  degree,  whereas  the  last  two  intersect  in  a 
curve  of  the  ninth  degree.  Hence  in  every  case  the  given  cubic  is  only  a  part  of 
the  curve  of  intersection  —  that  part  which  lies  on  all  three  cylinders. 

Again,  we  may  eliminate  u  from  equations  (i),  thus 

(ii)  xy  =  z,        y*  =  xz, 

of  which  the  first  defines  a  hyperbolic  paraboloid  and  the  second  a  hyperbolic- 
parabolic  cone.  The  straight  line  y  =  0,  z  =  0  lies  on  both  of  these  surfaces, 
but  not  on  the  cylinder  y  =  x2.  Hence  the  intersection  of  the  surfaces  (ii)  consists 
of  this  line  and  the  cubic.  The  generators  of  the  paraboloid  are  defined  by 

x  =  a,       z  =  ay ;        y  —  6,      z  =  bx ; 

for  all  values  of  the  constants  a  and  6.  From  (i)  we  see  that  the  cubic  meets  each 
generator  of  the  first  family  in  one  point  and  of  the  second  family  in  two  points. 

3.  Linear  element.  By  definition  the  length  of  an  arc  of  a  curve 
is  the  limit,  when  it  exists,  toward  which  the  perimeter  of  an 
inscribed  polygon  tends  as  the  number  of  sides  increases  and  their 
lengths  uniformly  approach  zero.  Curves  for  which  such  a  limit 
does  not  exist  will  be  excluded  from  the  subsequent  discussion. 

Consider  the  arc  of  a  curve  whose  end  points  m0,  ma,  are  deter 
mined  by  the  parametric  values  UQ  and  # ,  and  let  mv  m2,  •  •  • ,  be 
intermediate  points  with  parametric  values  u^  w2,  •  •  • .  The  length 
lk  of  the  chord  mkmk+l  is 

=V2,r/;.^,  L1)-/v(oi2.  « =  i,  2, 3 


LINEAR  ELEMENT  5 

By  the  mean  value  theorem  of  the  differential  calculus  this  is 
equal  to 


where  f  t.  =  wt  +  0<(w*+i  ~  %)»  °  <  ^  <  * 

and  the  primes  indicate  differentiation. 

,  As  denned,  the  length  of  the  arc  m0ma  is  the  limit  of  2Z4,  as  the 
lengths  inkmk+l  tend  to  zero.  From  the  definition  of  a  definite 
integral  this  limit  is  equal  to 


r a  n 


Hence,  if  s  denotes  the  length  of  the  arc  from  a  fixed  point  (u0) 
to  a  variable  point  (u),  we  have 


This  equation  gives  s  as  a  function  of  w.    We  write  it 
(12)  «=£(w), 

and  from  (11)  it  follows  that 


which  we  may  write  in  the  form 

(14)  ds2=dx'i 

As  thus  expressed  ds  is  called  the  element  of  length,  or  linear 
element,  of  the  curve. 

In  the  preceding  discussion  we  have  tacitly  assumed  that  u  is 
real.  When  it  is  complex  we  take  equation  (11)  as  the  definition 
of  the  length  of  the  arc. 

If  equation  (12)  be  solved  for  u  in  terms  of  s,  and  the  result 
be  substituted  in  (4),  the  resulting  equations  also  define  the  curve, 
and  s  is  the  parameter.  From  (11)  follows  the  theorem  : 

A  necessary  and  sufficient  condition  that  the  parameter  u  be  the 
arc  measured  from  the  point  U  =  UQ  is 

(15)  /,'2+/r+/s'2  =  l- 

An  exceptional  case  should  be  noted  here,  namely, 

/r+/2"+/a'2=o. 


6  CURVES  IN  SPACE 

Unless//, /2', /3'  be  zero  and  the  curve  reduce  to  a  point,  at  least  one 
of  the  coordinates  must  be  imaginary.  For  this  case  s  is  zero.  Hence 
these  imaginary  curves  are  called  curves  of  length  zero,  or  minimal 
curves.  For  the  present  they  will  be  excluded  from  the  discussion. 
Let  the  arc  be  the  parameter  of  a  given  curve  and  s  and  s  +  e 
its  values  for  two  points  M(x,  y,  z)  and  M^(x^  y^  z^.  By  Taylor's 
theorem  we  have 


(17) 


^  =  z  -f  z'e 


where  an  accent  indicates  differentiation  with  respect  to  s. 

Unless  x1,  y',  z!  are  all  zero,  that  is,  unless  the  locus  is  a  point 
and  not  a  curve,  one  at  least  of  the  lengths  xl—x,  y^—y,  zl—z  is 
of  the  order  of  magnitude  of  e.  If  these  lengths  be  denoted  by 
&u,  %,  Sz,  and  e  by  8«,  then  we  have 


where  12  denotes  the  aggregate  of  terms  of  the  second  and  higher 
orders  in  8s.  Hence,  as  Ml  approaches  M  the  ratio  of  the  lengths 
of  the  chord  and  the  arc  MMl  approaches  unity ;  and  in  the  limit 
we  have  ds2  =  dx2  +  dy2  +  dz2. 

4.  Tangent  to  a  curve.  The  tangent  to  a  curve  at  a  point  M  is 
the  limiting  position  of  the  secant  through  M  and  a  point  Ml  of 
the  curve  as  the  latter  approaches  M  as  a  limit. 

In  order  to  find  the  equation  of  the  tangent  we  take  s  for  par 
ameter  and  write  the  expressions  for  the  coordinates  of  M1  in  the 
form  (17).  The  equations  of  the  secant  through  M  and  Ml  are 


If  each  member  of  these  equations  be  multiplied  by  e  and  the 
denominators  be  replaced  by  their  values  from  (17),  we  have  in 
the  limit  as  M1  approaches  M 


y' 


TANGENT  TO  A  CURVE  7 

If  #,  /3,  7  denote  the  direction-cosines  of  the  tangent  in  conse 
quence  of  (15),  we  may  take 

When  the  parameter  u  is  any  whatever,  these  equations  are  * 

ft  f!  ft 

(20)  a=     /       Jl     =,  0=     .       /2     =  >  y  =  -=        3 

*  9  /  nt  *i  ^»  *  o     .        /wo  /  /»*o     .        /»*->.        /»!*)  *  // 


They  may  also  be  written  thus  : 

/oi\  dx       0      dy  dz 

21)  a  =  :T'     £  =  •/'     V  =  T" 

ds  ds  ds 

From  these  equations  it  follows  that,  if  the  convention  be  made 
that  the  positive  direction  on  the  curve  is  that  in  which  the  par 
ameter  increases,  the  positive  direction  upon  the  tangent  is  the 
same  as  upon  the  curve. 

A  fundamental  property  of  the  tangent  is  discovered  by  con 
sidering  the  expression  for  the  distance  from  the  point  M^  with 
the  coordinates  (17),  to  any  line  through  M.  We  write  the  equa 
tion  of  such  a  line  in  the  form 

(22)  *=£  =  !=*  =  £=£, 

a  b  c 

where  a,  5,  c  are  the  direction-cosines. 

The  distance  from  Ml  to  this  line  is  equal  to 

(23)  {[(bx>-  ay')e  +  \(bx"-  ay")e*  +  •  •  •  ]2 

bz')e  +  •  •  -]2+  [(az1-  cx')e  +  •  .  -]2}*. 


Hence,  if  MMl  be  considered  an  infinitesimal  of  the  first  order, 
this  distance  also  is  of  the  first  order  unless 


in  which  case  it  is  of  the  second  order  at  least.  But  when  these 
equations  are  satisfied,  equations  (22)  define  the  tangent  at  M. 
Therefore,  of  all  the  lines  through  a  point  of  a  curve  the  tangent 
is  nearest  to  the  curve. 

*  Whenever  the  functions  x',  y',  z'  appear  in  a  formula  it  is  understood  that  the  arc  s  is 
the  parameter  ;  otherwise  we  use  /{,  /2'  ,  /3'  ,  indicating  by  accents  derivatives  with  respect 
to  the  argument  u. 


8  CURVES  IN  SPACE 

5.  Order  of  contact.    Normal  plane.    When  the  curve  is  such 
that  there  are  points  for  which 

(24)  •    ^=4 

x'      y'      z' 

the  distance  from  Ml  to  the  tangent  is  of  the  third  order  at  least. 
In  this  case  the  tangent  is  said  to  have  contact  of  the  second  order, 
whereas,  ordinarily,  the  contact  is  of  the  first  order.  And,  in  gen 
eral,  the  tangent  to  a  curve  has  contact  of  the  wth  order  at  a  point, 
if  the  following  conditions  are  satisfied  for  n  =  2,  •  •  • ,  n  —  1,  and  n : 


xw 


i(«0 


jyV'V  5>V"y 

<25)  ^=^=^rr 

When  the  parameter  of  the  curve  is  any  whatever,  equations 
(24),  (25)  are  reducible  to  the  respective  equations 

fl  ~jTf  //    '  -f(.n-\)  f (rt-1)  f(n-l)' 

Jl  J%  J%  J\  J '2,  Ja 

The  plane  normal  to  the  tangent  to  a  curve  at  the  point  of 
contact  is  called  the  normal  plane  at  the  point.  Its  equation  is 

(26)  (X  —  x)  a  +  (Y  —  y}  ft  +  (Z  —  z)  7  =  0, 

where  a,  /3,  7  have  the  values  (20). 

EXAMPLES 

1.  Put  the  equations  of  the  circular  helix  (3)  in  the  form  (8). 

2.  Express  the  equations  of  the  circular  helix  in  terms  of  the  arc  measured  from 
a  point  of  the  curve,  and  show  that  the  tangents  to  the  curve  meet  the  elements  of 
the  circular  cylinder  under  constant  angle. 

3.  Show  that  if  at  every  point  of  a  curve  the  tangency  is  of  the  second  order, 
the  curve  is  a  straight  line. 

4.  Prove  that  a  necessary  and  sufficient  condition  that  at  the  point  (x0,  2/o)  of 
the  plane  curve  y  —f(x)  the  tangent  has  contact  of  the  nth  order  is/"(x0)  =  /'"(BO) 
=  . . .  r=/(«)(z0)  =  0  ;  also,  that  according  as  n  is  even  or  odd  the  tangent  crosses  the 
curve  at  the  point  or  does  not. 

5.  Prove  the  following  properties  of  the  twisted  cubic  : 

(a)  Of  all  the  planes  through  a  point  of  the  cubic  one  and  only  one  meets  the 
cubic  in  three  coincident  points ;  its  equation  is  3  u*x  -  3  uy  +  z  -  w3  =  0. 

(6)  There  are  no  double  points,  but  the  orthogonal  projection  on  a  plane  has  a 
double  point. 

(c)  Four  planes  determined  by  a  variable  chord  of  the  cubic  and  by  each  of 
four  fixed  points  of  the  curve  are  in  constant  cross-ratio. 


FIKST  CUKVATUKE  9 

6.  Curvature.  Radius  of  first  curvature.  Let  Jf,  Ml  be  two 
points  of  a  curve,  As  the  length  of  the  arc  between  these  points, 
and  A0  the  angle  between  the  tangents.  The  limiting  value  of 
A0/As  as  Ml  approaches  Jf,  namely  dd/ds,  measures  the  rate  of 
change  of  the  direction  of  the  tangent  at  M  as  the  point  of  con 
tact  moves  along  the  curve.  This  limiting  value  is  called  the 
first  curvature  of  the  curve  at  M,  and  its  reciprocal  the  radius  of 
first  curvature ;  the  latter  will  be  denoted  by  p. 

In  order  to  find  an  expression  for  p  in  terms  of  the  quantities 
defining  the  curve,  we  introduce  the  idea  of  spherical  representa 
tion  as  follows.  We  take  the  sphere  *  of  unit  radius  with  center 
at  the  origin  and  draw  radii  parallel  to  the  positive  directions  of- 
the  tangents  to  the  curve,  or  such  a  portion  of  it  that  no  two 
tangents  are  parallel.  The  locus  of  the  extremities  is  a  curve 
upon  the  sphere,  which  is  in  one-to-one  correspondence  with  the 
given  curve.  In  this  sense  we  have  a  spherical  representation,  or 
spherical  indicatrix,  of  the  curve. 

The  angle  A#  between  the  tangents  to  the  curve  at  the  points 
M,  M^  is  measured  by  the  arc  of  the  great  circle  between  their 
representative  points  m,  ml  on  the  sphere.  If  ACT  denotes  the 
length  of  the  arc  of  the  spherical  indicatrix  between  m  and  m^ 
then  by  the  result  at  the  close  of  §  3, 

dO      v     A<9 

—  =  lim  —  =  1. 

da  Ao- 

Hence  we  have 

(27)  !  =  £, 

p      ds 

where  da-  is  the  linear  element  of  the  spherical  indicatrix. 

The  coordinates  of  m  are  the  direction-cosines  a,  /3,  7  of  the 
tangent  at  M\  consequently 

When  the  arc  s  is  the  parameter,  this  formula  becomes 

(28)  ^.jji+yw+gw 

*  Hereafter  we  refer  to  this  as  the  unit  sphere. 


10  CURVES  IN  SPACE 

However,  when  the   parameter  is   any  whatever,   u,   we  have 
from  (12),  (13),  (20), 


and  W-fifl+JSJf*flfr 

Hence  we  find  by  substitution 

(30) 


which  sometimes  is  written  thus  : 

\  _(d*x}* 
f- 

The  sign  of  p  is  not  determined  by  these  formulas.  We  make 
the  convention  that  it  is  always  positive  and  thus  fix  the  sense  of 
a  displacement  on  the  spherical  indicatrix. 

7.  Osculating  plane.  Consider  the  plane  through  the  tangent  to 
a  curve  at  a  point  M  and  through  a  point  M^  of  the  curve.  The 
limiting  position  of  this  plane  as  Ml  approaches  M  is  called  the 
osculating  plane  at  M.  In  deriving  its  equation  and  thus  establish 
ing  its  existence  we  assume  that  the  arc  s  is  the  parameter,  and 
take  the  coordinates  of  Ml  in  the  form  (17). 

The  equation  of  a  plane  through  M  (x,  y,  z)  is  of  the  form 

(32)  (X-  x)a  +  (Y-  y)l  +  (Z-z)c  =  Q, 

X,  Y",  Z  being  the  current  coordinates.     When  the  plane  passes 
through  the  tangent  at  Jf,  the  coefficients  a,  £>,  c  are  such  that 

(33)  x'a  +  y'b  +  z'c  =  0. 

If  the  values  (17)  for  a;  ,  y^  z^  be  substituted  in  (32)  for  X,  F,  Z, 

e* 
and  the  resulting  equation  be  divided  by  —  ,  we  get 


where  77  represents  the  aggregate  of  the  terms  of  first  and  higher 
orders  in  e.    As  M1  approaches  Jf,  77  approaches  zero,  and  in  the 
limit  we  have 
(34)  x"a  +  y"b  +  z"c  =  0. 


OSCULATING  PLANE 


11 


Eliminating  a,  £>,  c  from  equations  (32),  (33),  (34)  we  obtain,  as 
the  equation  of  the  osculating  plane, 


(35) 


X-x 

x1 
x" 


Y-y 

y' 
y" 


Z-z 


=  0. 


From  this  we  find  that  when  the  curve  is  defined  by  equations  (4) 
in  terms  of  a  general  parameter  w,  the  equation  of  the  osculating 
plane  is  x_x  Y 

(36) 


y 


z_z 


//' 


/,"     •  fi 


The  plane  defined  by  either  of  these  equations  is  unique  except 
when  the  tangent  at  the  point  has  contact  of  an  order  higher  than 
the  first.  In  the  latter  case  equations  (33),  (34)  are  not  independent, 
as  follows  from  (24);  and  if  the  contact  of  the  tangent  is  of  the  wth 
order,  the  equations  ^  +  ^  +  ^G  =  0? 

for  all  values  of  r  up  to  and  including  n  are  not  independent  of 
one  another.  But  for  r  =  n  +  ~\.,  this  equation  and  (33)  are  inde 
pendent,  and  we  have  as  the  equation  of  the  osculating  plane  at  this 


singular  point, 


X-x 

x' 


Y- 
y' 


Z-z 


=  0. 


When  a  curve  is  plane,  and  its  plane  is  taken  for  the  rry-plane, 
the  equation  (35)  reduces  to  Z  =  0.  Hence  the  osculating  plane 
of  a  plane  curve  is  the  plane  of  the  latter,  and  consequently  is  the 
same  for  all  points  of  the  curve.  Conversely,  when  the  osculating 
plane  of  a  curve  is  the  same  for  all  its  points,  the  curve  is  plane, 
for  all  the  points  of  the  curve  lie  in  the  fixed  osculating  plane. 

The  equation  of  the  osculating  plane  of  the  twisted  cubic  (§2)  is  readily 
reducible  to 


where  JT,  F,  Z  are  current  coordinates.  From  the  definition  of  the  osculating  plane 
and  the  fact  that  the  curve  is  a  cubic,  it  follows  that  the  osculating  plane  meets 
the  curve  only  at  the  point  of  osculation.  As  equation  (i)  is  a  cubic  in  w,  it  follows 
that  through  a  point  («o,  2/o,  ZQ)  not  on  tne  cnrve  there  pass  three  planes  which 
osculate  the  cubic.  Let  MI,  w2,  u3  denote  the  parameter  values  of  these  points. 
Then  from  (i)  we  have 


=  3  XG, 


—  3  ?/o, 


2t  i  \  n 


12  CURVES  IN  SPACE 

By  means  of  these  relations  the  equation  of  the  plane  through  the  corresponding 
three  points  on  the  cubic  is  reducible  to 

(X  -  XQ)  3  7/0  -  (  Y  -  y0)  3  x0  +  (Z  -  z0)  =  0. 

This  plane  passes  through  the  point  (x0,  2/0,  ZQ)  5  hence  we  have  the  theorems  : 

The  points  of  contact  of  the  three  osculating  planes  of  a  twisted  cubic  through  a 
point  not  on  the  curve  lie  in  a  plane  through  the  point. 

The  osculating  planes  at  three  points  of  a  twisted  cubic  meet  in  a  point  which  lies 
•in  the  plane  of  the  three  points. 

By  means  of  these  theorems  we  can  establish  a  dual  relation  in  space  by  mak 
ing  a  point  correspond  to  the  plane  through  the  points  of  osculation  of  the  three 
osculating  planes  through  the  point,  and  a  plane  to  the  point  of  intersection  of  the 
three  planes  which  osculate  the  cubic  at  the  points  where  it  is  met  by  the  plane. 
In  particular,  to  a  point  on  the  cubic  corresponds  the  osculating  plane  at  the  point, 
and  vice  versa. 

8.  Principal  normal  and  binormal.  Evidently  there  are  an  in 
finity  of  normals  to  a  curve  at  a  point.  Two  of  these  are  of  par 
ticular  interest  :  the  normal,  which  lies  in  the  osculating  plane  at 
the  point,  called  the  principal  normal;  and  the  normal,  which  is 
perpendicular  to  this  plane,  called  the  binormal. 

If  the  direction-cosines  of  the  binormal  be  denoted  by  X,  />t,  z>, 
we  have  from  (35) 


X  :  /»  :  v  =  (y'z"-  z'y")  :  (z'x"~  z'z")  :  (*'/'-  y'x"). 
In  consequence  of  the  identity 


the  value  of  the  common  ratio  is  reducible  by  means  of  (19)  and 
(28)  to  ±p.*  We  take  the  positive  direction  of  the  binormal  to 
be  such  that  this  ratio  shall"  be  -f-  p  ;  then 

(37)       \  =  p(yfz"-z'y"),     ^  =  P(z'x"~  x'z"),     v  =  p(x'y» 
When  the  parameter  u  is  general,  these  formulas  are 


(38)  x=282 

or  in  other  form  : 

/oof  dycPz  —  dzcfy  dzd*x—dxd?z  dxd'y  —  dyd^x 

~P       ~~df~       '  ^~P~        ds3  ~P  ds* 

*For  S»V/=  0,  as  is  seen  by  differentiating  2x"*=  1  with  respect  to  s. 


PRINCIPAL  NORMAL  AND  BINORMAL 


13 


By  definition  the  principal  normal  is  perpendicular  to  both  the 
tangent  and  binormal.  We  make  the  convention  that  its  positive 
direction  is  such  that  the  positive  directions  of  the  tangent,  prin 
cipal  normal  and  binormal  at  a  point  have  the  same  mutual  ori 
entation  as  the  positive  directions 
of  the  x-,  y-,  z-axes  respectively. 
These  directions  are  represented  in 
fig.  2  by  the  lines  MT,  MC,  MB. 
Hence,  if  Z,  m,  n,  denote  the  direc 
tion-cosines  of  the  principal  normal, 
we  have* 


(39) 


m 

/JL 


=4-1, 


FIG.  2 


from  which  it  follows  that 

I  a  =  mv  —  nu,      ft  —  n\  —  Iv,      7  =  Z/i  —  m\, 
I  =  fjij  —  vft^      m  =  va  —  \7,     n  =  Xp  —  yuo:, 
\  =  ftn  —  777*,      /Ji  =  yl  —  #n,      i^  =  am  —  ftl. 
, 

Substituting  the  values  of  #,  /3,  7;  X,  /u-,  i^  from  (19)  and  (37)  in  the 
expressions  for  £,  m,  w,  the  resulting  equations  are  reducible  to 

Hence,  when  the  parameter  u  is  general,  we  have 


(42)   l=-(W- 
or  in  other  form, 


_ 


22  —  dzd2s 


In  consequence  of  (29)  equations  (42)  may  be  written: 

da  dft  dj 

ds '  ds  ds' 

or  by  means  of  (27), 

da  dft  dy 


(43) 


m 


do-  da- 


Hence  the  tangent  to  the  spherical  indicatrix  of  a  curve  is  parallel 
to  the  principal  normal  to  the  curve  and  has  the  same  sense. 

*C.  Smith,  Solid  Geometry,  llth  ed.,  p.  31. 


14  CURVES  IN  SPACE 

9.  Osculating  circle.  Center  of  first  curvature.  We  have  defined 
the  osculating  plane  to  a  curve  at  a  point  M  to  be  the  limiting 
position  of  the  plane  determined  by  the  tangent  at  M  and  by  a 
point  Ml  of  the  curve,  as  the  latter  approaches  M  along  the  curve. 
We  consider  now  the  circle  in  this  plane  which  has  the  same  tan 
gent  at  M  as  the  curve,  and  passes  through  M{.  The  limiting  posi 
tion  of  this  circle,  as  Ml  approaches  7I/,  is  called  the  osculating  circle 
to  the  curve  at  M.  It  is  evident  that  its  center  C0  is  on  the  prin 
cipal  normal  at  M.  Hence,  with  reference  to  any  fixed  axes  in  space, 
the  coordinates  of  (70,  denoted  by  XQ,  F0,  ZQ,  are  of  the  form 

X0=x  +  rl,     Y0  —  y  -f  rm,     Z^=z  +  rn, 

where  the  absolute  value  of  r  is  the  radius  of  the  osculating  circle. 
In  order  to  find  the  value  of  r,  we  return  to  the  consideration 
of  the  circle,  when  Ml  does  not  have  its  limiting  position,  and  we 
let  X,  F,  Z\  Zj,  m^  n^  r^  denote  respectively  coordinates  of  the  cen 
ter  of  the  circle,  the  direction-cosines  of  the  diameter  through  M 
and  the  radius.  If  xv  yv  zl  be  the  coordinates  of  M^  they  have  the 
values  (17),  and  since  Ml  is  on  the  circle,  we  have 

rl  =  2(A'-  xtf  =  2(7-^-  ex'-  i  e*x".  •  -)2. 


If  we  notice  that  ^x'^  =  0,  and  after  reducing  the  above  equation 
divide  through  by  e2,  we  have 

1-r^Z/'  +*?  =  (), 

where  77  involves  terms  of  the  first  and  higher  orders  in  e.    In  the 

limi-t  rl  becomes  r,  ^x\  becomes  2z'7,  that  is  -  ,  and  this  equation 
reduces  to 


so  that  r  is  equal  to  the  radius  of  curvature.  On  this  account  the 
osculating  circle  is  called  the  circle  of  curvature  and  its  center  the 
center  of  first  curvature  for  the  point.  Since  r  is  positive  the  center 
of  curvature  is  on  the  positive  half  of  the  principal  normal,  and 
consequently  its  coordinates  are 

(44)  X0=x  +  pl,     Y0=y  +  pm,     ZQ=  z  +  pn. 


CENTER  OF  CURVATURE  15 

The  line  normal  to  the  osculating  plane  at  the  center  of  curva 
ture  is  called  the  polar  line  or  polar  of  the  curve  for  the  corre 
sponding  point.  Its  equations  are 

/45\  X-x-pl  =  Y-y-pm  =  Z—z  —  pn  ^ 

\  JJL  v 

In  fig.  2  C  represents  the  center  of  curvature  and  CP  the  polar 
line  for  M. 

A  curve  may  be  looked  upon  as  the  path  of  a  point  moving  under  the  action  of 
a  system  of  forces.  From  this  point  of  view  it  is  convenient  to  take  for  parameter 
the  time  which  has  elapsed  since  the  point  passed  a  given  position.  Let  t  denote  this 
parameter.  As  t  is  a  function  of  8,  we  have 

dx  _dx  ds  _     ds      dy  _    ds      dz  _     ds 
¥~  ds~dt~  C*dt'     dt~    ~dt      ~dt~y~dt' 

Hence  the  rate  of  change  of  the  position  of  the  point  with  the  time,  or  its  velocity, 

may  be  represented  by  the  length  —  laid  off  on  the  tangent  to  the  curve.    In  like 

dt 
manner,  by  means  of  (41),  we  have 

d^  _     d?s      n  /ds\2 
~7 


From  this  it  is  seen  that  the  rate  of  change  of  the  velocity  at  a  point,  or  the 
acceleration,  may  be  represented  by  a  vector  in  the  osculating  plane  at  the  point, 
through  the  latter  and  whose  components  on  the  tangent  and  principal  normal 


d*s  ,  1  /d§Y 
— -  and  -  I  — ) 
df*  P  \dtj 


EXAMPLES 

1.  Prove  that  the  curvature  of  a  plane  curve  defined  by  the  equation  M (x,  y)dx 

cy       ex 


p  (J/2  +  N 

2.  Show  that  the  normal  planes  to  the  curve, 

x  —  a  sin2  it,     y  =  a  sin  u  cos  w,     z  =  a  cos  M, 
pass  through  the  origin,  and  find  the  spherical  indicatrix  of  the  curve. 

3.  The  straight  line  is  the  only  real  curve  of  zero  curvature  at  every  point. 

4.  Derive  the  following  properties  of  the  twisted  cubic : 

(a)  In  any  plane  there  is  one  line,  and  only  one,  through  which  two  osculating 
planes  can  be  drawn. 

(6)  Four  fixed  osculating  planes  are  cut  by  the  line  of  intersection  of  any  two 
osculating  planes  in  four  points  whose  cross-ratio  is  constant. 

(c)  Four  planes  through  a  variable  tangent  and  four  fixed  points  of  the  curve 
are  in  constant  cross-ratio. 

(d)  What  is  the  dual  of  (c)  by  the  results  of  §  7  ? 


16  CURVES  IN  SPACE 

5.  Determine  the  form  of  the  function  0  so  that  the  principal  normals  to  the 
curve  x  =  w,  y  =  sin  w,  z  -  <f>  (u)  are  parallel  to  the  yz-plane. 

6.  Find  the  osculating  plane  and  radius  of  first  curvature  of 

x  —  a  cos  u  -f  6  sin  w,     y  =  a  sin  u  +  6  cos  w,     z  =  c  sin  2  u. 

10.  Torsion.  Frenet-Serret  formulas.  It  has  been  seen  that,  un 
less  a  curve  be  plane,  the  osculating  plane  varies  as  the  point 
moves  along  the  curve.  The  change  in  the  direction  depends 
evidently  upon  the  form  of  the  curve.  The  ratio  of  the  angle  A^ 
between  the  binormals  at  two  points  of  the  curve  and  their  curvi 
linear  distance  As  expresses  our  idea  of  the  mean  change  in  the 
direction  of  the  osculating  plane.  And  so  we  take  the  limit  of 
this  ratio,  as  one  point  approaches  the  other,  as  the  measure  of 
the  rate  of  this  change  at  the  latter  point.  This  limit  is  called 
the  second  curvature,  or  torsion,  of  the  curve,  and  its  inverse  the 
radius  of  second  curvature,  or  the  radius  of  torsion.  The  latter 
will  be  denoted  by  r. 

In  order  to  establish  the  existence  of  this  limit  and  to  find  an 
expression  for  it  in  terms  of  the  functions  defining  the  curve, 
we  draw  radii  of  the  unit  sphere  parallel  to  the  positive  binormals 
of  the  curve  and  take  the  locus  of  the  end  points  of  these  radii  as 
a  second  spherical  representation  of  the  curve.  The  coordinates  of 
points  of  this  representative  curve  on  the  sphere  are  X,  /*,  v.  Pro 
ceeding  in  a  manner  similar  to  that  in  §  6,  we  obtain  the  equation 

(46)  i_£ 

r2      ds* 

where  dcrl  is  the  linear  element  of  the  spherical  indicatrix  of  the 
binormals. 

In  order  that  a  real  curve  have  zero  torsion  at  every  point,  the  cosines  X,  /*,  v 
must  be  constant.  By  a  change  of  the  fixed  axes,  which  evidently  has  no  effect 
upon  the  form  of  the  curve,  the  cosines  can  be  given  the  values  X  =  1,  /*  =  v  =  0. 
It  follows  from  (40)  that  a  =  0,  and  consequently  x  —  const.  Hence  a  necessary 
and  sufficient  condition  that  the  torsion  of  a  real  curve  be  zero  at  every  point  is 
that  the  curve  be  plane. 

In  the  subsequent  discussion  we  shall  need  the  derivatives  with 
respect  to  s  of  the  direction-cosines  a,  &  7;  I,  m,  w;  X,  p,  v.  We 
deduce  them  now.  From  (41)  we  have 

(4T)  a'=i,     /3'=™,     y-». 


FRENET-SERRET  FORMULAS 


17 


In  order  to  find  the  values  of  X',  /*',  i/,  we  differentiate  with 
respect  to  s  the  identities, 

X2+At2+z,2  =  1?  a\  +  £/A  +  7*  =  0, 

and,  in  consequence  of  (47),  obtain 

XX'  +  fjLfj,1  +  vv1  =  0,          «\'  +  /V  +  yvr  =  0. 
From  these,  by  (40),  follows  the  proportion 

X':  fjLf:vr=l:m:n, 

and  the  factor  of  proportionality  is  ±  1/r,  as  is  seen  from  (46). 
The  algebraic  sign  of  r  is  not  determined  by  the  latter  equation. 
We  fix  its  sign  by  writing  the  above  proportion  thus : 


(48) 


V  ,:'-  „'- 

A  =  —  •>       l*>  =  —  t       V  =•  — 

T  T  T 


If  the  identity  I  =  ^7  —  vfi  be  differentiated  with  respect  to  s 
the  result  is  reducible  by  (40),  (47),  and  (48)  to 

(49)  I— 

Similar  expressions  can  be  found  for  m'  and  n'.  Gathering  to 
gether  these  results,  we  have  the  following  formulas  fundamental  in 
the  theory  of  twisted  curves,  and  called  the  Frenet-Serret  formulas : 


(50) 


*-i,  / 

*-5, 

y'= 

W 

—  » 

p 

p 

/> 

l'=-(- 

+-Y 

«'  =  • 

-(-- 

^ 

\p 

T7 

V 

T 

%,    I 

,_m 

/ 

n 

X  =  —  j      / 

"  — 

—  • 

T 

T 

T 

£Y   ^-/l+2), 

v  \^    v 


As  an  example,  we  derive  another  expression  for  the  torsion. 
If  the  equation  \  =  p(y'z"—  z'y") 

be  differentiated  with  respect  to  s,  the  result  may  be  written 


If  this  equation  and  similar  ones  for  WI/T,  rz/r  be  multiplied  by  ?,  wz, 
n  respectively  and  added,  we  have,  in  consequence  of  (50)  and  (41), 


x' 


(51) 


y 

y" 


z' 

z" 


x'"     y'"     z'" 


18  CUKVES  IN  SPACE 

The  last  three  of  equations  (50)  give  the  rate  of  change  of  the 
direction-cosines  of  the  osculating  plane  of  a  curve  as  the  point  of 
osculation  moves  along  the  curve.  From  these  equations  it  follows 
that  a  necessary  and  sufficient  condition  that  this  rate  of  change 
at  a  point  be  zero  is  that  the  values  of  s  for  the  point  make  the 
determinant  in  equation  (51)  vanish.  At  such  a  point  the  osculat 
ing  plane  is  said  to  be  stationary. 

11.  Form  of  curve  in  the  neighborhood  of  a  point.  The  sign  of 
torsion.  We  have  made  the  convention  that  the  positive  directions 
of  the  tangent,  principal  normal,  and  binormal  shall  have  the  same 
relative  orientation  as  the  fixed  x-,  y-,  2-axes  respectively.  When  we 
take  these  lines  at  a  point  MQ  for  axes,  the  equations  of  the  curve 
can  be  put  in  a  very  convenient  form.  If  the  coordinates  be  ex 
pressed  in  terms  of  the  arc  measured  from  M^  we  have  from  (19) 
and  (41)  that  for  s  =  0 

P 

When  the  values  of  I  and  X  from  (41)  and  (37)  are  substituted  in 
the  fourth  of  equations  (50),  we  obtain 

(5 2)  x'"  =  ----  (y'z" - z'y") - £-' x". 

p        r   '  p 

From  this  and  similar  expressions  for  y'"  and  z"1  we  find  that 
for  s  =  0  -i  f  -i 

P*  p2  PT 

Hence,  by  Maclauriri's  theorem,  the  coordinates  #,  y,  z  can  be  ex 
pressed  in  the  form  /  -i 

(^%\  <  v  = —  s3-f---, 

\<JO  I  v      •      Ct  £J'2 

2  p      b  p 

z  —     —      s  -f-  •  •  • , 
6  pr 

where  p  and  r  are  the  radii  of  first  and  second  curvature  at  the 
point  s  =  0,  and  the  unwritten  terms  are  of  the  fourth  and  higher 
powers  in  s. 

From  the  last  of  these  equations  it  is  seen  that  for  sufficiently 
small  values  of  8  the  sign  of  z  changes  with  the  sign  of  s  unless 


THE  SIGN  OF  TORSION  19 

I/T  =  0  at  M0.  Hence,  unless  the  osculating  plane  is  stationary  at 
a  point,  the  curve  crosses  the  plane  at  the  point.*  Furthermore, 
when  a  point  moves  along  a  curve  in  the  positive  direction,  it 
passes  from  the  positive  to  the  negative  side  of  the  osculating 
plane  at  a  point,  or  vice  versa,  according  as  the  torsion  at  the 
latter  is  positive  or  negative.  In  the  former  case  the  curve  is 
said  to  be  sinistrorsum,  in  the  latter  dextrorsum. 

As  another  consequence  of  this  equation,  we  remark  that  as  a 
variable  point  M  on  the  curve  approaches  Jf0,  the  distance  from  M 
to  the  osculating  plane  at  MQ  is  of  the  third  order  of  magnitude  in 
comparison  with  MMQ.  By  means  of  the  other  equations  (53)  we 
find  that  the  distance  to  any  other  plane  through  M0  is  of  the 
second  order  at  most.  Hence  we  have  the  theorem : 

The  osculating  plane  to  a  twisted  curve  at  an  ordinary  point  is 
crossed  by  the  curve,  and  of  all  the  planes  through  the  point  it  lies 
nearest  to  the  curve. 

From  the  second  of  (53)  it  is  seen  that  y  is  positive  for  suffi 
ciently  small  values  of  «,  positive  or  negative.  Hence,  in  the 
neighborhood  of  an  ordinary  point,  the  curve  lies  entirely  on  one 
side  of  the  plane  determined  by  the  tangent  and  binomial  —  on 
the  side  of  the  positive  direction  of  the  principal  normal. 

These  properties  of  a  twisted  curve  are  discovered,  likewise, 
from  a  consideration  of  the  projections  upon  the  coordinate  planes 
of  the  approximate  curve,  whose  equations  consist  of  the  first 
terms  in  (53).  The  projection  on  the  osculating  plane  is  the 
parabola  x  =  *,  y  =  s2/2  p,  whose  axis  is  the  principal  normal 
to  the  curve.  On  the  plane  of  the  tangent  and  binomial  it  is 
the  cubic  x  =  s,  z  =  —  s8/6  pr,  which  has  the  tangent  to  the 
curve  for  an  inflectional  tangent.  And  the  curve  projects  upon 
the  plane  of  the  binormal  and  principal  normal  into  the  semi- 
cubical  parabola  y  =  s2/2  /o,  z=  —  s3/Q  pr,  with  the  latter  for 
cuspidal  tangent. 

These  results  are  represented  by  the  following  figures,  which  picture  the  pro 
jection  of  the  curve  upon  the  osculating  plane,  normal  plane,  and  the  plane  of  the 
tangent  and  binormal.  In  the  third  figure  the  heavy  line  corresponds  to  the  case 
where  r  is  positive  and  the  dotted  line  to  the  case  where  r  is  negative. 

*This  result  can  be  derived  readily  by  geometrical  considerations. 


20 


CUEVES  IN  SPACE 


The  preceding  results  serve  also  to  give  a  means  of  determining;  ue  variation  in 
the  osculating  plane  as  the  point  moves  along  the  curve.  By  r  ns  of  (50)  the 
direction-cosines  X,  /u,,  v  can  be  given  the  form 


where  the  subscript  null  indicates  the  value  of  a  function  for  s  =  0  and  the  un 
written  terms  are  of  the  second  and  higher  terms  in  s.  If  the  coordinate  axes  are 
those  which  lead  to  (53),  the  values  of  X,  p,  v  for  the  point  of  parameter  5s  are 

X  =  0         =-,     v  =  \ 

TO' 

to  within  terms  of  higher  order,  and  consequently  the  equation  of  this  osculating 
plane  at  this  point  MI  is  «. 

Y-  +  Z  =  0. 
TO 

If  we  put  Y  =  po,  we  get  the  z-coordinate  of  the  point  in  which  this  plane  is  cut  by 
the  polar  line  for  the  point  s  =  0  ;  it  is  —  po5s/T0 .  Hence,  according  as  TO  is  positive 
or  negative  at  Jf,  the  osculating  plane  at  the  near-by  point  MI  cuts  the  polar  line  for 
M  on  the  negative  or  positive  side  of  the  osculating  plane  at  M . 

12.  Cylindrical  helices. 

As  another  example  of  the  use  of  formulas  (50)  we  derive  several  properties  of 
cylindrical  helices.  By  definition,  a  cylindrical  helix  is  a  curve  which  lies  upon  a 
cylinder  and  cuts  the  elements  of  the  cylinder  under  constant  angle.  If  the  axis  of 
z  be  taken  parallel  to  the  elements  of  the  cylinder,  we  have  7  =  const.  Hence, 
from  (50), 


from  which  it  follows  that  the  cylindrical  helices  have  the  following  properties : 
The  principal  normal  is  perpendicular  to  the  element  of  the  cylinder  at  the  point, 

and  consequently  coincides  with  the  normal  to  the  cylinder  at  the  point  (§  22). 
The  radii  of  first  and  second  curvature  are  in  constant  ratio. 
Bertrand  has  established  the  converse  theorem  :  Every  curve  whose  radii  of  first 

and  second  curvature  are  in  constant  ratio  is  a  cylindrical  helix.    In  order  to  prove 

it,  we  put  T  =  icp,  and  remark  from  (50)  that 

dv 


da_d\ 
ds         ds 


dp  _    dfj.      dy  _ 
ds~     ds '     ds        ds 


from  which  we  get       a  —  K\  +  a,     /3  =  */*  +  6,     7  =  KV  +  c, 
where  a,  6,  c  are  constants.    From  these  equations  we  find 

a2  +  52  +  C2  _  i  +  K^        aa  +  bp  +  cy  = 


CYLINDRICAL  HELICES  21 

Hence  the  ta    -ents  to  the  curve  make  the  constant  angle  cos-  *  — =;  with  the 
lines  whose  lin      3n-cosines  are — V  *         Consequently  the  curve  is  a  cylindrical 

Vl    +   K2 

helix,  and  t'.u  e  ts  of  the  helix  have  the  above  direction. 


EXAMPLES 

1.  Find  the  length  of  the  curve  x  =  a  (u  —  sin  w),  y  =  a  cos  w,  between  the  points 
for  which  u  has  the  values  —  TT  and  IT  ;  show  that  the  locus  of  the  center  of  curva 
ture  is  of  the  same  form  as  the  given  curve. 

2.  Find  the  coordinates  of  the  center  of  curvature  of 

x  =  a  cos  M,     y  =  a  sin  w,     z  —  a  cos  2  u. 

3.  Find  the  radii  of  curvature  and  torsion  of 

x  =  a  (u  —  sinw),    y  =  a  (1  —  cosw),     z  =  bu. 

4.  If  the  principal  normals  of  a  curve  are  parallel  to  a  fixed  plane,  the  curve 
is  a  cylindrical  helix. 

5.  Show  that  the  curve  x  —  eu,  y  =  er  M,  z  —  V%  u  is  a  cylindrical  helix  and  that 
the  right  section  of  the  cylinder  is  a  catenary  ;  also  that  the  curve  lies  upon  a  cylin 
der  whose  right  section  is  an  equilateral  hyperbola.  Express  the  coordinates  in  terms 
of  the  arc  and  find  the  radii  of  first  and  second  curvature. 

6.  Show  that  if  6  and  0  denote  the  angles  which  the  tangent  and  binormal  to  a 

sin  6  dd       r 

curve  make  with  a  fixed  line  in  space,  then  -  =  -  - 

sin  </>  d<p      p 

7.  When  two  curves  are  symmetric  with  respect  to  the  origin,  their  radii  of 
first  curvature  are  equal  and  their  radii  of  torsion  differ  only  in  sign. 

8.  The  osculating  circle  at  an  ordinary  point  of  a  curve  has  contact  of  the  sec 
ond  order  with  the  latter  ;  and  all  other  circles  which  lie  in  the  osculating  plane 
and  are  tangent  to  the  curve  at  the  point  have  contact  of  the  first  order. 

9.  A  necessary  and  sufficient  condition  that  the  osculating  circle  at  a  point  have 
contact  of  the  third  order  is  that  p'  =  0  and  I/T  =  0  at  the  point  ;  at  such  a  point 
the  circle  is  said  to  superosculate  the  curve. 

10.  Show  that  any  twisted  curve  may  be  defined  by  equations  of  the  form 


where  p  and  r  are  the  radii  of  first  and  second  curvature  at  the  point  s  —  0. 
11.  When  the  equations  of  a  curve  are  in  the  form  (4),  the  torsion  is  given  by 

/I       /2       /3 
f       fff       f/f 

f  £ 

where  0  has  the  significance  of  equation  (12). 


22  CURVES  IN  SPACE 

12.  The  locus  of  the  centers  of  curvature  of  a  twisted  curve  of  constant  first 
curvature  is  a  curve  of  the  same  kind. 

13.  When  all  the  osculating  planes  of  a  curve  pass  through  a  fixed  point,  the 
curve  is  plane. 

14.  Determine  f(u)  so  that  the  curve  x  =  a  cosw,  y  =  a  sin  w,  z  =f(u)  shall  be 
plane.    What  is  the  form  of  the  curve  ? 

13.  Intrinsic  equations.  Fundamental  theorem.  Let  C^  and  Cz  be 
two  curves  defined  in  terms  of  their  respective  arcs  s,  and  let  points 
upon  each  with  the  same  values  of  s  correspond.  We  assume, 
furthermore,  that  at  corresponding  points  the  radii  of  first  curva 
ture  have  the  same  value,  and  also  the  radii  of  second  curvature. 
We  shall  show  that  Cl  and  Cz  are  congruent. 

By  a  motion  in  space  the  points  of  the  two  curves  for  which 
s  =  0  can  be  made  to  coincide  in  such  a  way  that  the  tangents, 
principal  normals,  and  binomials  to  them  at  the  point  coincide 
also.  Hence  if  we  use  the  notation  of  the  preceding  sections  and 
indicate  by  subscripts  1  and  2  the  functions  of  Cl  and  C2,  we  have, 
when  s  =  0, 

(54)  xl  =  xz,     al  =  az,     ^  =  Z2,     \  =  \z, 

and  other  similar  equations. 

The  Frenet-Serret  formulas  for  the  two  curves  are 


ds  ds  r          ds 


—  =—  =  —  >     —  =  —  (  —  -|  --  I  »     -=-  =  —  -> 
ds      p       ds         \p       T/        ds       r 

the  functions  without  subscripts  being  the  same  for  both  curves. 
If  the  equations  of  the  first  row  be  multiplied  by  «2,  Z2,  X2  respec 
tively,  and  of  the  second  row  by  a^  l^  X:,  and  all  added,  we  have 

(55)  ^(«,«.H*,+  xi\)=0. 

and  consequently  a^  +  IJ2  +  \\  =  const. 

This  constant  is  equal  to  unity  for  s  =  0,  as  is  seen  from  (54),  and 
hence  for  all  values  of  s  we  have 


INTRINSIC  EQUATIONS  23 

Combining  this  equation  with  the  identities 


we  obtain  (at  -  a2)2  +  ft  -  £>)  2  +  (X,  -  X2)2  =  0. 

Hence  al  =  a#  ^  =  Z2,  Xt  =  X2.    Moreover,  since  in  like  manner 

&  =  &'  7i  =  72»  we  nave 

1(^-^=0,    ^-^=0,    I  <*,-«,)  =  o. 

Consequently  the  differences  2^—  #2,  y^—y^  zl  —  z2  are  constant. 
But  for  s  =  0  they  are  zero,  and  so  we  have  the  theorem  : 

Two  curves  whose  radii  of  first  and  second  curvature  are  the  same 
functions  of  the  arc  are  congruent. 

From  this  it  follows  that  a  curve  is  determined,  to  within  its 
position  in  space,  by  the  expressions  for  the  radii  of  first  and  second 
curvature  in  terms  of  the  arc.  And  so  the  equations  of  a  curve 
may  be  written  in  the  form 

(56)  />=/,<«),        T  =/,(.). 

They  are  called  its  intrinsic  equations. 

We  inquire,  conversely,  whether  two  equations  (56),  in  which  f^ 
and/2  are  any  functions  whatever  of  a  parameter  s,  are  intrinsic 
equations  of  a  curve  for  which  s  is  the  length  of  arc. 

In  answering  this  question  we  show,  in  the  first  place,  that  the 
equations 

/trrv  du      v       dv          /u      w\       dw      v 

(pi)  —  -  =  -,     —  =  —  /--|  —  ,  —  __ 

ds      p       ds          \p       T/       ds      r 

admit  of  three  sets  of  solutions,  namely  : 

(58)  u  =  a,  v  =  1,  w  =  \;  u  =  fi,  v  =  m,  w  =  /JL  ;  u  —  y^  v  =  n,  w  =  v; 

which  are  such  that  for  each  value  of  s  the  quantities  a,  fi,  7; 
/,  7?z,  n  ;  X,  /-i,  v  are  the  direction-cosines  of  three  mutually  perpen 
dicular  lines.  In  fact,  we  know  *  that  a  system  (57)  admits  of  a 
unique  set  of  solutions  whose  values  for  s  =  0  are  given  arbitra 
rily.  Consequently  these  equations  admit  of  three  sets  of  solutions 


*  Picard,  Tralte  d' Analyse,  Vol.  II,  p.  313;  Goursat,  Cours  d' Analyse  Mathematique, 
Vol.  II,  p.  356. 


24  CURVES  IN  SPACE 

whose  values  for  s  =  0  are  1,  0,  0 ;  0,  1,  0 ;  0,  0,  1  respectively. 
By  an  argument  similar  to  that  applied  to  equation  (55)  we  prove 
that  for  all  values  of  s  the  solutions  (58)  satisfy  the  conditions 

(59)  aft  +  Im  +  \p  =  0,     £7  +  mn  +  pv  =  0,     ya  +  nl  +  v\  =  0. 
In  like  manner,  since  it  follows  from  (57)  that 

du        dv         dw      A 
u-r+v  —  +  w  —  =  (), 
as         as          as 

we  prove  that  these  solutions  satisfy  the  conditions 

(60)  oa+Za+Xa  =  l,     /32  +  m2+/x2  =  l,     y+n'+i/^l. 

But  the  conditions  (59),  (60)  are  equivalent  to  (40),  and  conse 
quently  the  three  sets  of  functions  a,  &  7;  Z,  TH,  w;  X,  /z,  v  are 
the  direction-cosines  of  three  mutually  perpendicular  lines  for  all 
values  of  s. 

Suppose  we  have  such  a  set  of  solutions.    For  the  curve 

(61)  x—  I  ads,     y=  I  fids,     z=*  I  yd*, 

the  functions  a,  /3,  7  are  the  direction-cosines  of  the  tangent,  and 
since  ds*  =  dx2  +  dy*  +  dz2,  s  measures  the  arc  of  the  curve.  From 
(61)  and  the  first  of  (57)  we  get 

d?x_l_      ofy^m      d^z^n.     /d*x\*     /^>\2     /^\2=  1 
ds*~p'     ds2~  p*     df~~p'     W/      W/      W/      p*' 

Hence  if  p  be  positive  for  all  values  of  s,  it  is  the  radius  of  curva 
ture  of  the  curve  (61),  and  Z,  m,  n  are  the  direction-cosines  of  the 
principal  normal  in  the  positive  sense.  In  consequence  of  (40)  the 
functions  X,  yu.,  v  are  the  direction-cosines  of  the  binomial;  hence 
from  (50)  and  the  third  of  (57)  it  follows  that  r  is  the  radius  of 
torsion  of  the  curve.  Therefore  we  have  the  following  theorem 
fundamental  in  the  theory  of  curves : 

Given  any  two  analytic  functions,  f^s),  f2(s),  of  which  the  former 
is  positive  for  all  values  of  s  within  a  certain  domain  ;  there  exists  a 
curve  for  which  p  =/j(s),  r  =/2(«),  and  s  is  the  arc,  for  values  of  s  in 
the  given  domain.  The  determination  of  the  curve  reduces  to  the  find 
ing  of  three  sets  of  solutions  of  equations  (57),  satisfying  the  conditions 
(59),  (60),  and  to  quadratures. 


KICCATI  EQUATIONS  25 

We  proceed  now  to  the  integration  of  equations  (57).   Since  each 
set  of  integrals  of  the  desired  kind  must  satisfy  the  relation 


(62)  u2 

we  introduce  with  Darboux  *  two  functions  cr  and  &),  defined  by 


(63) 


1  —  w      u  —  iv 
u  —  iv      1  +  w 


\  —  w      u  +  iv          ft) 

It  is  evident  that  the  functions  cr  and  —  -  are  conjugate  imaginaries. 
Solving  for  u,  v,  w,  we  get 

1  —  o-ft)  ,1  +  o-ft)  cr  +  co 

(64)  u  = 1     v  =  i i     w  = 


If  these  values  be  substituted  in  equations  (57),  it  is  found  that 
the  functions  cr  and  co  are  solutions  of  the  equation 

(65)  Miieip. 

ds      2  r      p         2  r 

And  conversely,  any  two  different  solutions  of  (65),  when  substi 
tuted  in  (64),  lead  to  a  set  of  solutions  of  equations  (57)  satisfying 
the  relation  (62).  Our  problem  reduces  then  to  the  integration  of 
equation  (65). 

14.    Riccati  equations.    Equation  (65)  may  be  written 

(66)  ^  =  L  +  2  MO  +  NP, 

where  L,  M,  N  are  functions  of  s.  This  equation  is  a  generalized 
form  of  an  equation  first  studied  by  Riccati,  f  and  consequently 
is  named  for  him.  As  Riccati  equations  occur  frequently  in  the 
theory  of  curves  and  surfaces,  we  shall  establish  several  of  their 
properties. 

Theorem.  When  a  particular  integral  of  a  Riccati  equation  is 
known,  the  general  integral  can  be  obtained  by  two  quadratures. 

*  Lemons  sur  la  Thdorie  Generate  des  Surfaces,  Vol.  I,  p.  22.  We  shall  refer  to  this 
treatise  frequently,  and  for  brevity  give  our  references  the  form  Darboux,  I,  22. 

t  Cf .  Forsyth,  Differential  Equations,  chap,  v ;  also  Cohen,  Differential  Equations, 
pp.  173-177. 


26  CURVES  IN  SPACE 

Let  Ol  be  a  particular  integral  of  (66).    If  we  put  0  =  l/c£  -f  0^ 
the  equation  for  the  determination  of  </>  is 

(67)  -£  +  2(M+2WJ<l>+N**Q. 

As  this  equation  is  linear  and  of  the  first  order,  it  can  be  solved  by 
two  quadratures.  Since  the  general  integral  of  (67)  is  of  the  form 
<£=/1(s)+  0/2(3),  where  a  denotes  the  constant  of  integration,  the 
general  integral  of  equation  (66)  is  of  the  form 

«        •  -Sri- 

where  P,  (),  R,  S  are  functions  of  s. 

Theorem.  When  two  particular  integrals  of  a  Riccati  equation  are 
known,  the  general  integral  can  be  found  by  one  quadrature. 

Let  0l  and  02  be  two  solutions  of  equation  (66).    If  we  effect  the 
substitution  6  —  --  \-6  ,  the  equation  in  ^  is 


If  this  equation  and  (67)  be  multiplied  by  1/^r  and  !/<£  respec 
tively,    and    subtracted,   the    resulting    equation   is    reducible    to 

—  ty/<l>)=N(01—02)^/<t>.    Consequently  the  general  integral  of 
cts 

(66)  is  given  by 

0—0,        -v|r  fv(0t-0c/« 

<69> 


where  a  is  the  constant  of  integration. 

Since  equation  (68)  may  be  looked  upon  as  a  linear  fractional 
substitution  upon  a,  four  particular  solutions  0V  02,  #3,  04,  corre 
sponding  to  four  values  av  az,  a3,  a±  of  a,  are  in  the  same  cross-ratio 
as  these  constants.  Hence  we  have  the  theorem  : 

The  cross-ratio  of  any  four  particular  integrals  of  a  Riccati 
equation  is  constant. 

From  this  it  follows  that  if  three  particular  integrals  are  known, 
the  general  integral  can  be  obtained  without  quadrature. 


DETERMINATION  OF  COORDINATES 


27 


15.    The  determination  of  the  coordinates  of  a  curve  defined  by 
its  intrinsic  equations.    We  return  to  the  consideration  of  equation 

(65)  and  indicate  by 

a.P+O  bP+Q 

(70)  o\  =  -^ — -^i        (oi  =  T—> — -v  (i  =  l,2,3) 

six  particular  integrals  of  this  equation.    From  these  we  obtain 
three  sets  of  solutions  of  equations  (57),  namely 


(71) 


*1      I 


and  similar  expressions  in  cr2,  &>2 ;  <73,  &>3  respectively  for  /8,  m,  ft; 
7,  n,  ZA  These  expressions  satisfy  the  conditions  (60).  In  order 
that  (59)  also  may  be  satisfied  we  must  have 


CT—  ft),,       ft),— ft), 


which  is  reducible  to 
(72) 


=  -1. 


z.  _  1     9    q  *  ^ 

A/  JL  ^     — ^     O 

5X ;  «2,  J2 ;  a3, 


Hence  each  two  of  the  three  pairs  of  constants 
form  a  harmonic  range. 

When  the  values  (70)  for  <T.,  &)t.  are  substituted  in  the  expressions 
for  a,  /3,  7,  it  is  found  that 


(73) 


7  = 


,1+a, 


a  — 


where,  for  the  sake  of  brevity,  we  have  put 


(74) 


RS  —  PQ 
PS-QR 


28  CURVES  IN  SPACE 

The  coefficients  of  U,  F,  and  W  in  (73)  are  of  the  same  form  as 
the  expressions  (71)  for  cr,  Z,  X  ;  £,  ra,  /*  ;  7,  n,  z>.  Moreover,  the 
equations  of  condition  (59)  are  equivalent  to  (72).  Hence  these 
coefficients  are  the  direction-cosines  of  three  fixed  directions  in 
space  mutually  perpendicular  to  one  another.  If  lines  through  the 
origin  of  coordinates  parallel  to  these  three  lines  be  taken  for  a  new 
set  of  axes,  the  expressions  for  #,  /3,  7  with  reference  to  these  axes 
reduce  to  U,  V,  ^respectively.*  These  results  may  be  stated  thus : 

If  the  general  solution  of  equation  (65)  be 

(68)  "=STV 

aii  -f-  o 

the  curve  whose  radii  of  first  and  second  curvature  are  p  and  T 
respectively  is  given  by 


f\  T)v  »  I/ 

w-K  ) 

/~r>  ci 
RS-- 
PS- 


QR 

It  must  be  remarked  that  the  new  axes  of  coordinates  are  not 
necessarily  real,  so  that  when  it  is  important  to  know  whether  the 
curves  are  real  it  will  be  advisable  to  consider  the  general  formulas 
(73).  An  example  of  this  will  be  given  later. 

We  shall  apply  the  preceding  results  to  several  problems. 

When  the  curve  is  plane  the  torsion  is  zero,  and  conversely.   For  this  case  equa 
tion  (65)  reduces  to  —  =  —  -  0  of  which  the  general  integral  is 
ds         p 

-if— 
0  =  ae   1J  P  =  ae~ i<r, 

where  a  is  an  arbitrary  constant,  and  by  (27)  <r  is  the  measure  of  the  arc  of  the 
spherical  indicatrix  of  the  tangent.    This  solution  is  of  the  form  (08),  with 

O    -I 


Therefore  the  coordinates  are  given  by 

(75)  x=Ccos<rds,     y=Csir\<rds,     2  =  0 

Hence  the  coordinates  of  any  plane  curve  can  be  put  in  this  form. 

*  This  is  the  same  thing  as  taking 

ai  —  —  &i  =  l,     c*2=—  b^— i,    «3  — oo,     &3=0. 

t  Scheffers,  Anwendung  der  Differential  und  Integral  Rechnung  auf  Geometrie, 
Vol.  I,  p.  219.    Leipsic,  1902. 


DETERMINATION  OF  COORDINATES  29 

We  have  seon  that  cylindrical  helices  are  characterized  by  the  property  that  the 
radii  of  first  und  second  curvature  are  in  constant  ratio.  If  we  put  T  =  pc,  equation 
(65)  may  bj  written 

^  =  _l(l_2c0-02). 
ds      2  T  V 

Two  particular  integrals  are  the  roots  of  the  equation  02  +  2  cd  -  1  =  0.    These 
roots  are  real  and  unequal  if  c  is  real ;  we  consider  only  this  case,  and  put 


(76)  el  =  -c  -Vc2  +  1,     02  =  -  c  +  Vc2  +  l,     01&2  =  -  1. 

From  (69)  it  follows  that  the  general  solution  of  the  above  equation  is 

(77) 

where  we  have  put 

(78) 

Since  <r  and  --  in  (63)  are  conjugate  imaginary,  if  we  take 


then  a  and  6  must  be  such  that 


aeit  _  i  &oe-  my  _  0j 

where  60  denotes  the  conjugate  imaginary  of  6.    This  reduces,  in  consequence  of 

(76)' to  n  4-  M 

(79,  ;        ,      *  =  -£±|=-* 

One  solution  of  this  is  given  by  taking  <x>  and  0  for  a  and  6;  we  put  as  =  GO, 
63  =  0.  If  these  values  be  substituted  in  (72),  we  get  a»  4-  k  -  0,  where  i  =  1,  2.  So 
that  equation  (79)  becomes  &&•  0  =  0^,  where  i  =  1,  2.  The  solutions  of  this  equation 
are  61  =  01?  62  =  -  i0i-  From  (77)  P  =  e«02,  Q  =  -  0i,  R  =  e»'r,  S  =  -  1,  so  that 

W=—=- 
Vc2  +  l 

When  the  foregoing  values  are  substituted  in  (73),  and  the  resulting  values  of 
a,  /3,  7  in  (61),  we  get 

(80)  x-        C    -  Ccoslds,     y=  -   fstafdt,     g  =  =  • 

Vc2  +  1  J  Vc2  +  1  J  Vc2  +  1 

From  the  la  ie  expressions  we  find  that  the  tangent  to  the  curve  makes 

a  constant  ang  c  -v\        the  z-axis  —  the  direction  of  the  elements  of  the  cylinder. 
And  the  cross-section  of  the  cylinder  is  defined  by 


Xi  =  fcos  t  dsi,       y\  =  J  sin  t  dsi, 


where  Si  denotes  the  arc  of  this  section  measured  from  a  point  of  it.    If  pi  denotes 
the  radius  of  curvature  of  the  right  section,  we  find  that  pc2  =  pi(c2  +  !)• 


30  CURVES  IN  SPACE 

EXAMPLES 

1.  Find  the  coordinates  of  the  cylindrical  helix  whose  intrinsic  equations  are 

p  =   T   =  S. 

2.  Show  that  the  helix  whose  intrinsic  equations  are  p  =  T  =  (s2  +  4)/V2  lies 
upon  a  cylinder  whose  cross-section  is  a  catenary. 

3.  Establish  the  following  properties  for  the  curve  with  the  intrinsic  equations 
p  =  as,  r  =  6s,  where  a  and  b  are  constants : 

(a)  the  Cartesian  coordinates  are  reducible  to  x=Aeht  cos£,  y  =  Aehtsm  t,  z  —  Behi, 
where  J_,  B,  h  are  functions  of  a  and  6 ; 

(6)  the  curve  lies  upon  a  circular  cone  whose  axis  coincides  with  the  z-axis  and 
cuts  the  elements  of  the  cone  under  constant  angle. 

16.  Moving  trihedral.  In  §  11  we  took  for  fixed  axes  of  refer 
ence  the  tangent,  principal  normal,  and  binormal  to  a  curve  at  a 
point  MQ  of  it,  and  expressed  the  coordinates  of  any  other  point  of 
the  curve  with  respect  to  these  axes  as  power  series  in  the  arc  s 
of  the  curve  between  the  two  points.  Since  M0  is  any  point  of  the 
curve,  there  is  a  set  of  such  axes  for  each  of  its  points.  Hence, 
instead  of  considering  only  the  points  whose  locus  is  the  curve, 
we  may  look  upon  the  moving  point  as  the  intersection  of  three 
mutually  perpendicular  lines  which  move  along  with  the  point, 
the  whole  figure  rotating  so  that  in  each  position  the  lines  coin 
cide  with  the  tangent,  principal  normal,  and  binormal  at  the  point. 
We  shall  refer  to  such  a  configuration  as  the  moving  trihedral. 
In  the  solution  of  certain  problems  it  is  of  advantage  to  refer  the 
curve  to  this  moving  trihedral  as  axes.  We  proceed  to  the  con 
sideration  of  this  idea. 

With  reference  to  the  trihedral  at  a  point  Jf,  the  direction- 
cosines  of  the  tangent,  principal  normal,  and  binormal  at  M 
have  the  values 

a=l,   /3  =  7  =  0;     I  =  0,  m=l,  n  =  0;     X  =  p  =  0,  i/  =  l. 

As  the  trihedral  begins  to  move,  the  rates  of  change  of  these 
functions  with  s  are  found  from  the  Frenet  formulas  (50)  to 
have  the  values 

da  _  ft    d{$  _  1     dy  _  ft  dl  _       1     dm  _  ~ 

— —  =  o    ——  =  — ,  — —  —  o  j          — -  —         5  —    —  u, 
ds  ds      p     ds  ds          p     ds 

dn          1        d\      n     dfJL      1     dv      ft 

—  = j       — r~  =  "i     ~T~  =  ~  '      7~  =  U* 

ds          r        ds  ds      r     ds 


MOVING  TRIHEDRAL 


31 


Let  f,  77,  f  denote  coordinates  referring  to  the  axes  at  Jf,  and 
f,  77',  f  those  with  reference  to  the  axes  at  Jf',  and  let  JfJf'  =  As 
(see  fig.  4).  Since  the  rate  of  change  of  a  is  zero  and  a  =1  at  Jf, 
the  cosine  of  the  angle  between  the  £-  and  f'-axes  is  1  to  within 
terms  of  higher  than  the  first  order  in  As.  Likewise  the  cosine  of 
the  angle  between  the  f-  and  Tj'-axes  is  —  As/p.  We  calculate  the 
cosines  of  the  angles  between  all 
the  axes,  and  the  results  may  be 
tabulated  as  follows: 

f        v       S 


(81) 


As 


.,       _  As 


FIG.  4 


Let  P  be  a  point  whose  coordinates  with  respect  to  the  trihedral 
at  M  are  f,  77,  f.  Suppose  that  as  Jf  describes  the  given  curve  (7, 
P  describes  a  path  T.  It  may  happen  that  in  this  motion  P  is  fixed 
relatively  to  the  moving  trihedral,  but  in  general  the  change  in  the 
position  of  P  will  be  due  not  only  to  the  motion  of  the  trihedral 
but  also  to  a  motion  relative  to  it.  In  the  latter  general  case,  if  P' 
denotes  the  point  on  T  corresponding  to  M'  on  (7,  the  coordinates 
of  P'  relative  to  the  axes  at  M  and  M  '  may  be  written 

?4Af»  Tj+A^,   f+A^;     f  +  A2f,  17  +A^,  f+A2?. 
Thus  A2#  indicates  the  variation  of  a  function  0  relative  to  the 
moving  trihedral,  and  A^  the  variation  due  to  the  latter  and  to 
the  motion  of  the  trihedral. 

To  within  terms  of  higher  order  the  coordinates  of  M'  are 
(As,  0,  0)  with  respect  to  the  axes  at  Jf,  and  with  the  aid  of  (81) 
the  equations  of  the  transformation  of  coordinates  with  respect 
to  the  two  axes  are  expressible  thus  : 


. 

— 


32  CURVES  IN  SPACE 

These  reduce  to 


As       As  o  As       As 


.  .     . 

H 


As       As  /a  T 

In  the  limit  as  Jtf  '  approaches  M  these  equations  become 


ds      c?s  p       ds      ds      p      T      ds      ds      T 

80  d0 

thus  -T-  denotes  the  absolute  rate  of   change  of  0,  and  -=-  that 

relative  to  the  trihedral.* 

If  t  denotes  the  distance  between  P  and  a  point  P^  (f^  rj^  fj), 
that  is  ^2  =  (?1-f)2+(7?1-7;)2  +  (?'1-?)2,  we  find  by  means  of  the 
formulas  (82)  that 


If  a,  5,  <?  denote  the  direction-cosines  of  PP^  with  respect  to 
the  axes  at  Jf,  then 


When  we  express  the  condition  that  f1?  77  L,  ft  as  well  as  f,  ?/,  f 
satisfy  equations  (82),  we  are  brought  to  the  following  fundamental 
relations  between  the  variations  of  a,  6,  c: 

&a      da      b        &b      db      a      c         $c  __  dc      b 

(83)          ~T~  ~  ~7  ---  '      T"  ==  ~T  —  I  ---  1  —  *      ~r  ==  "T         * 
c?s      ds      p        ds      ds      p      T        ds      ds      r 

If  the  point  P  remains  fixed  in  space  as  M  moves  along  the 
curve,  the  left-hand  members  of  equations  (82)  are  zero  and 
the  equations  reduce  to 


(84)  --l,  = 

ds      p                 ds          \p      T/  c?s      T 

Moreover,  the  direction-cosines  of  a  line  fixed  in  space  satisfy 

the  equations 

or.            -      da      b           db          (a      c\  dc      b 

(85)  -—  =  -»         —  =  —  (-  +  -  )i  _  =  _. 
c?s      /)           c?s          \p      T)  ds      r 

*  Cf.  Cesaro,  Lezioni  di  Geometria  Intrinseca,  pp.  122-128.    Naples,  1896. 


MOVING  TRIHEDRAL  33 

These  are  the  Frenet-Serret  formulas,  as  might  have  been 
expected. 

We  shall  show  that  the  solution  of  these  equations  carries  with 
it  the  solution  of  (84).  Suppose  we  have  three  sets  of  solutions 
of  (85),  a,  I,  X  ;  &  m,  /JL  ;  7,  n,  v,  whose  values  for  s  =  0  are 

(86)  1,     0,     0;          0,     1,     0;         0,     0,     1. 

They  are  the  direction-cosines,  with  respect  to  the  moving  trihedral 
with  vertex  M,  of  three  fixed  directions  in  space  mutually  perpen 
dicular  to  one  another.  Let  0  be  a  fixed  point,  and  through  it 
draw  the  lines  with  the  directions  just  found.  Take  these  lines 
for  coordinate  axes  and  let  #,  y,  z  denote  the  coordinates  of  M 
with  respect  to  them.  If  f,  ??,  f  denote  the  coordinates  of  0 
with  respect  to  the  moving  trihedral,  then  —  f  ,  —  ?;,  —  f  are  the 
coordinates  of  M  with  respect  to  the  trihedral  with  vertex  at 
0  and  edges  parallel  to  the  corresponding  edges  of  the  trihedral 
at  M.  Consequently  we  have 

f  =  -  (ax  -f  &y  +  7*)> 

(87)  rj=-(lx+my  +  m), 


If  these  values  be  substituted  in  (84)  and  we  take  account  of 
(50)  and  (85),  we  find  that  the  equations  are  identically  satisfied. 
If  fo»  ^o'  ?o  Denote  the  values  of  f,  ?;,  £  for  s  =  0,  it  follows  from 
(86)  and  (87)  that  they  differ  only  in  sign  from  the  initial  values 
of  x,  y,  z.  Hence  if  we  write,  in  conformity  with  (21), 

(88)       x 

and  substitute  these  values  in  (87),  they  become  the  general  solu 
tion  of  equations  (84).  We  have  seen  that  the  solution  of  equa 
tions  (85)  reduces  to  the  integration  of  the  Riccati  equation  (65). 

17.  Illustrative  examples.  As  an  example  of  the  foregoing  method  we  consider 
the  curve  which  is  the  locus  of  a  point  on  the  tangent  to  a  twisted  curve  C  at  a 
constant  distance  a  from  the  point  of  contact. 

The  coordinates  of  the  point  MI  of  the  curve  with  reference  to  the  axes  at  M 
are  a,  0,  0.  In  this  case  equations  (82)  reduce  to 

(i)  ^-l^-a^-0 

ds~    '      ds~  P'      ds~ 


34  CURVES  IN  SPACE 

Hence  if  Si  denotes  the  length  of  arc  of  C\  from  the  point  corresponding  to  s  =  0 
on  C,  we  have 


and  the  direction-cosines  of  the  tangent  to  Ci  with  reference  to  the  moving  axes 
are    iven  b 


are  given  by 


7a2  +  p2  Va2  4-  p2 


Hence  the  tangent  to  Ci  is  parallel  to  the  osculating  plane  at  the  corresponding 
point  of  C. 


point 

By  means  of  (83)  we  find 

d  /        p       \_          a 


Sa:\       d  / 
ds       ds  y  V(i2 


+  p2        p  Va2  +  p2      («2  +  P2)       P  Va2  +  p2 


Proceeding  in  like  manner  with  0i  and  71,  and  making  use  of  (ii),  we  have 
d(*i  &2pp'  d  5/3i         —  ftp'2p'  p 


_ 

(a2  +  p'2)'2      a'2  +  p2  '  5Si  ~  (a2  +  p2)2      a2  +  p2' 

871  _  ap 


From  these  expressions  and  (21'}  we  obtain  the  following  expression  for  the  square 
of  the  first  curvature  of  C\ : 

app/    -II  -t 


Pi2      a2  +  p2  \a2  + 
The  direction-cosines  of  the  principal  normal  of  C\  are 

5/?i  571 


By  means  of  (40)  we  derive  the  following  expressions  for  the  direction-cosines  of 
the  binormal : 

r  (a2  +  p2)^  r  (a2  +  p2)^  >        ft2  +  ?'  Va2  +  p2 

In  order  to  find  the  expression  for  TI,  the  radius  of  torsion  of  Ci,  we  have  only  to 
substitute  the  above  values  in  the  equation 


_  S\i  _         p         /d\i  _  MI\ 
~  5«i      Va2  +  p2  \  ds       P' 


We  leave  this  calculation  to  the  reader  and  proceed  to  an  application  of  the 
preceding  results. 

We  inquire  whether  there  is  a  curve  C  such  that  Ci  is  a  straight  line.  The 
necessary  and  sufficient  condition  is  that  I/pi,  be  zero  (Ex.  3,  p.  15).  From  (iii)  it 
follows  that  we  must  have 


ILLUSTRATIVE  EXAMPLES  35 

From  the  second  of  these  equations  it  follows  that  C  must  be  plane,  and  from 
the  former  we  get,  by  integration, 

log  (a* +  ,*)  =  —  +  «, 

where  c  is  a  constant  of  integration.    If  the  point  s  =  0  be  chosen  so  that  we  may 
take  c  =  log  a2,  this  equation  reduces  to 


=  a  \e  «  -  1. 


P 
If  6  denotes  the  angle  which  the  line  C\  makes  with  the  £-axis,  we  have,  from  (i), 

8rj      a  1 

tan  6  =  —  =  -  = 


e«  -  1 
Differentiating  this  equation  with  respect  to  s,  we  can  put  the  result  in  the  form 

dd__  1 

ds          p ' 
consequently 

(89) 


When  these  values  are  substituted  in  equations  (75),  we  obtain  the  coordinates  of  C 
in  the  form 


x  =   i    \  1  —  e    «  as,  y  =  ae 

or,  in  terms  of  0, 
(90)  x  =  —  a    log  tan  -  +  cos  6  ,  y  =  a  sin  0. 

The  curve,  with  these  equations,  is  called  the  tractrix.  As  just  seen,  it  possesses 
the  property  that  there  is  associated  with  it  a  straight  line  such  that  the  segments 
of  the  tangents  between  the  points  of  tangency  and  points  of  intersection  with  the 
given  line  are  of  constant  length. 

Theorem.  The  orthogonal  trajectories  of  the  osculating  plane  of  a  twisted  curve  can 
be  found  by  quadratures. 

With  reference  to  the  moving  axes  the  coordinates  of  a  point  in  the  osculating 
plane  are  (£,  77,  0).  The  necessary  and  sufficient  condition  that  this  point  describe 
an  orthogonal  trajectory  of  the  osculating  plane  as  M  moves  along  the  given  curve 

is  that  —  and  —  in  (82)  be  zero.    Hence  we  have  for  the  determination  of  £  and  77 
ds         ds 

the  equations  ?*_„+         0  ^  +  £  =  0, 

da-  da- 

where  a  is  given  by  (89).    Eliminating  £,  we  have 

d?-t] 

^  +  "  =  '- 

Hence  77  can  be  found  by  quadratures  as  a  function  of  <r,  and  consequently  of  S, 
and  then  £  is  given  directly. 


36  CURVES  IK  SPACE 

Problem.  Find  a  necessary  and  sufficient  condition  that  a  curve  lie  upon  a  sphere. 

If  £,  ??,  f  denote  the  coordinates  of  the  center,  and  R  the  radius  of  the  sphere, 
we  have  £2  -f  if*  +  f 2  =  R2.  Since  the  center  is  fixed,  the  derivatives  of  £,  17,  f  are 
given  by  (84).  Consequently,  when  we  differentiate  the  above  equation,  the  result 
ing  equation  reduces  to  £  =  0,  which  shows  that  the  normal  plane  to  the  curve 
at  each  point  passes  through  the  center  of  the  sphere.  If  this  equation  be  differen 
tiated,  we  get  77  =  p ;  hence  the  center  of  the  sphere  is  on  the  polar  line  for  each 
point.  Another  differentiation  gives,  together  with  the  preceding,  the  following 
coordinates  of  the  center  of  the  sphere : 


When  the  last  of  these  equations  is  differentiated  we  obtain  the  desired  condition 
(92)  -  -f  (rp'Y  =  0. 

Conversely,  when  this  condition  is  satisfied,  the  point  with  the  coordinates  (91)  is 
fixed  in  space  and  at  constant  distance  from  points  of  the  curve.  A  curve  which 
lies  upon  a  sphere  is  called  a  spherical  curve.  Hence  equation  (92)  is  a  necessary 
and  sufficient  condition  that  a  curve  be  spherical. 

EXAMPLES 

1.  Let  C  be  a  plane  curve  and  Ci  an  orthogonal  trajectory  of  the  normals  to  C. 
Show  that  the  segments  of  these  normals  between  C  and  Ci  are  of  the  same  length. 

2.  Let  C  and  Ci  be  two  curves  in  the  same  plane,  and  say  that  the  points  corre 
spond  in  which  the  curves  are  met  by  a  line  through  a  fixed  point  P.    Show  that 
if  the  tangents  at  corresponding  points  are  parallel,  the  two  curves  are  similar 
and  P  is  the  center  of  similitude. 

3.  The  locus  of  the  point  of  projection  of  a  fixed  point  P  upon  the  tangent  to 
a  curve  C  is  called  the  pedal  curve  of  C  with  respect  to  P.   Show  that  if  r  is  the 
distance  from  P  to  a  point  M  on  (7,  and  6  the  angle  which  the  line  PM  makes 
with  the  tangent  to  C  at  M,  the  arc  Si  and  radius  of  curvature  pi  of  the  pedal 
curve  are  given  by 


where  s  and  p  are  the  arc  and  the  radius  of  curvature  of  C. 

4.  Find  the  intrinsic  and  parametric  equations  of  a  plane  curve  which  is  such 
that  the  segment  on  any  tangent  between  the  point  of  contact  and  the  projection 
of  a  fixed  point  is  of  constant  length. 

5.  Find  the  intrinsic  equation  of  the  plane  curve  which  meets  under  constant 
angle  all  the  lines  passing  through  a  fixed  point. 

6.  The  plane  curve  which  is  such  that  the  locus  of  the  mid-point  of  the  seg 
ment  of  the  normal  between  a  point  of  the  curve  and  the  center  of  curvature  is 
a  straight  line  is  the  cycloid  whose  intrinsic  equation  is  p2  -f-  s2  =  a2. 

7.  Investigate  the  curve  which  is  the  locus  of  the  point  on  the  principal  normal 
of  a  given  curve  and  at  constant  distance  from  the  latter. 


OSCULATING  SPHERE  37 

18.  Osculating  sphere.  Consider  any  curve  whatever  referred 
to  its  moving  trihedral.  The  point  whose  coordinates  have  the 
values  (91)  lies  on  the  normal  to  the  osculating  plane  at  the 
center  of  curvature,  that  is,  on  the  polar  line.  Consequently 
the  moving  sphere  whose  center  is  at  this  point,  and  whose 
radius  is  V p2  -f-  r2//2,  cuts  the  osculating  plane  in  the  osculating 
circle.  This  sphere  is  called  the  osculating  sphere  to  the  curve  at 
the  point.  We  shall  derive  the  property  of  this  sphere  which 
accounts  for  its  name. 

When  the  tangent  to  a  curve  at  a  point  M  is  tangent  likewise 
to  a  sphere  at  this  point,  the  center  of  the  sphere  lies  in  the  normal 
plane  to  the  curve  at  M.  If  R  denotes  its  radius  and  the  curve  is 
referred  to  the  trihedral  at  M,  the  coordinates  of  the  center  C  of  the 
sphere  are  of  the  form  (0,  yv  zt)  and  yl  +  z*  =  ^2.  Let  P(x,  y,  z) 
be  a  point  of  the  curve  near  M,  and  Q  the  point  in  which  the  line 
CP  cuts  the  sphere.  If  PQ  be  denoted  by  8,  we  have,  from  (53), 


6/r 
which  reduces  to 


Hence  8  is  of  the  second  order,  in  comparison  with  JMTP,  unless 
?/1=/3,  that  is,  unless  the  center  is  on  the  polar  line;  then  it  is 
of  the  third  order  unless  zl  =  —  p'r,  in  which  case  the  sphere  is 
the  osculating  sphere.  Hence  we  have  the  theorem : 

The  osculating  sphere  to  a  curve  at  a  point  has  contact  with  the 
curve  of  the  third  order ;  oilier  spheres  with  their  centers  on  the 
polar  line,  and  tangent  to  the  curve,  have  contact  with  the  curve  of 
the  second  order ;  all  other  spheres  tangent  to  the  curve  at  a  point 
have  contact  of  the  first  order. 

The  radius  of  the  osculating  sphere  is  given  by 

(93)  JS*  =,!»'+ TV, 

and  the  coordinates  of  the  center,  referred  to  fixed  axes  in  space,  are 

(94)  xl  =  x  +  pi  —  pfT\,     y^  =  y  +  pm  —  p'rfji,     zl  =  z  +  pn  —  p'rv. 


88 


CURVES  IN  SPACE 


Hence  when  p  is  constant  the  centers  of  the  osculating  sphere 
and  the  osculating  circle  coincide.    Then  the  radius  of  the  sphere  is 
necessarily  constant.    Conversely,  it  follows  from  the  equation  (93) 
IP          that  a  necessary  and  sufficient  condition  that  E  be  con 
stant  is 
[P, 


that  is,  either  the  curvature  is  constant,  or  the  curve 
is  spherical. 

If  equations  (94)  be  differentiated  with  respect  to  s, 

we  get 


(96)- 


#1  = 


From  these  expres 
sions  it  is  seen  that 
the  center  of  the 
osculating  sphere  is 
fixed  only  in  case 
of  spherical  curves. 
Also,  the  tangent  to 
the  locus  of  the  cen 
ter  is  parallel  to  the 
binormal.  Combin 
ing  this  result  with 
a  previous  one,  we 
have  the  theorem: 
The  polar  line  for  a  point  on  a  curve  is  tangent  to  the  locus  of  the 
center  of  the  osculating  sphere  to  the  curve  at  the  corresponding  point. 

This  result  is  represented  in  fig.  5,  in  which  the  curve  is  the 
locus  of  the  points  M\  the  points  (7,  C^  C2,  •  •  •  are  the  correspond 
ing  centers  of  curvature ;  the  planes  MCN,  M^C^N^  •  •  •  are  normal 
to  the  curve ;  the  lines  CP,  C^P^  •  •  •  are  the  polar  lines ;  and  the 
points  P,  Pj,  P2,  •  •  •  are  the  centers  of  the  osculating  spheres. 


FIG.  5 


BEETEAND  CUKVES  39 

19.  Bertrand  curves.  Bertrand  proposed  the  following  problem  : 
To  determine  the  curves  whose  principal  normals  are  the  principal 
normals  of  another  curve.  In  solving  this  problem  we  make  use  of 
the  moving  trihedral.  We  must  find  the  necessary  and  sufficient 
condition  that  the  point  Ml  ( £  =  0,  TJ  =  k,  ?  =  0)  generate  a  curve  C^ 
whose  principal  normal  coincides  with  the  ?;-axis  of  the  moving 
trihedral.  Since  the  point  M1  remains  on  the  moving  ?/-axis,  we 
have  d%  =  d%  =  0.  And  since  Ml  tends  to  move  at  right  angles  to 
this  axis,  Brj  =  0.  Now  equations  (82)  reduce  to 

(96)  }Ll-»,         £vO,          5--*. 

ds  p  ds    >  ds          r 

From  the  second  we  see  that  k  is  a  constant.  Moreover,  if  co  denotes 
the  angle  which  the  tangent  at  Ml  makes  with  the  tangent  at  M, 
we  have,  from  the  first  and  third  of  these  equations, 

8?  kp 

tan  co  =  -^r  = 


or  Sf      T  (k  —  p) 

sin  co      cos  co      sin  co 


(97) 


k 


We  have  seen  (§  11)  that  according  as  r  is  positive  or  negative, 
the  osculating  plane  to  a  curve  at  a  point  M'  near  M  cuts  the 
polar  line  for  M  below  or  above  the  osculating  plane  at  M.  From 
these  considerations  it  follows  that  when  r  >  0,  co  is  in  the  third, 
fourth,  or  first  quadrants  according  as  k  >  ^,  0  <  k  <  p,  or  k  <  0  ; 
and  when  r  <  0,  co  is  in  the  second,  first,  or  fourth  quadrant, 
accordingly.  It  is  readily  found  that  these  results  are  consistent 
with  equation  (97). 

By  means  of  (97)  it  is  found  from  (96)  that 


the  negative  sign  being  taken  so  that  the  left-hand  member  may 
be  positive. 

Thus  far  we  have  expressed  only  the  condition  that  the  locus 
of  M^  cut  the  moving  T^-axis  orthogonally,  but  not  that  this  axis 
shall  be  the  principal  normal  to  the  curve  Cl  also.  For  this  we 
consider  the  moving  trihedral  for  Cl  and  let  ax,  b^  c^  denote  the 


40 


CUKVES  IN  SPACE 


direction-cosines  with  respect  to  it  of  a  fixed  direction  in  space, 
as  M^D  in  fig.  6.    They  satisfy  equations  similar  to  (85),  namely 


M 


(99)       l 


If  a,  6,  c  are  the  direction-cosines  of  the 
same  direction,  with  respect  to  the  mov 
ing  trihedral  at  M,  we  must  have  al  —  a  cos  &>  -f-  c  sin  &), 
bl  =  6,  ^  =  —  a  sin  eo  +  <?  cos  a>,  for  all  possible  cases, 
as  enumerated  above.  When  these  values  are  sub 
stituted  in  the  above  equations,  we  get,  by  means 
of  (98), 


P 
sin  &) 


sin  tw 

_l_  _ 

T  p 

COS  ft) 


sm 


cos  ft)  —  a  sin  &))  — -  =  0, 
as 


[T  sin  G)      cos  <w      sin  &)1         Tsin  to      sin  &)      cos  &)1     _  ~ 
-          I  \d  -\-  I      -       I  (  I  c  —  u, 

*/>       PI       TI  J     L  *       PI       TI  J 

[sin  &)      cos  &)             k       ~| ,    ,    ,     .  x  dco       ~ 
. 6  -h  (<?  sm  &)  +  a  cos  &))—-  =  0. 
/3              T          rtr  sin  &)J  as 

Since  these  equations  must  be  true  for  every  fixed  line,  the  coeffi 
cients  of  a,  6,  c  in  each  of  these  equations  must  be  zero.  The 
resulting  equations  of  condition  reduce  to 


&)  =  const., 


(100) 


sm  &)      cos  &)      sm  <w      ~ 
1 ;; —  =  "• 


Since  &)  is  a  constant,  equation  (97)  is  a  linear  relation  between 
the  first  and  second  curvatures  of  the  curve  C.  And  the  last  of 
equations  (100)  shows  that  a  similar  relation  holds  for  the  curve  Cr 
Conversely,  given  a  curve  C  whose  first  and  second  curvatures 
satisfy  the  relation 

(ioi)  p+7  =  c> 

where  -4,  B,  C  are  constants  different  from  zero ;  if  we  take 

A  B 

k  =  —,  COt  ft)  = ;> 


TANGENT  SURFACE  OF  A  CURVE  41 

and  for  pl  and  rl  the  values  given  by  (100),  equations  (99)  are  sat 
isfied  identically,  and  the  point  (0,  k,  0)  on  the  principal  normal 
generates  the  curve  Cv  conjugate  to  C.  We  gather  these  results 
about  the  curves  of  Bertrand  into  the  following  theorem: 

A  necessary  and  sufficient  condition  that  the  principal  normals 
of  one  curve  be  the  principal  normals  of  a  second  is  that  a  linear 
relation  exist  between  the  first  and  second  curvatures;  the  distance 
between  corresponding  points  of  the  two  curves  is  constant,  the  oscu 
lating  planes  at  these  points  cut  under  constant  angle,  and  the  torsions 
of  the  two  curves  have  the  same  sign. 

We  consider,  finally,  several  particular  cases,  which  we  have 
excluded  in  the  consideration  of  equation  (101). 

When  C  =  0  and  A=£Q,  the  ratio  of  p  and  T  is  constant.  Hence 
the  curve  is  a  helix  and  its  conjugate  is  at  infinity.  When  A  =  0, 
that  is,  when  the  curve  has  constant  torsion,  the  conjugate  curve 
coincides  with  the  original.  When  A  =  C  =  0,  k  is  indeterminate  ; 
hence  plane  curves  admit  of  an  infinity  of  conjugates,  —  they  are 
the  curves  parallel  to  the  given  curve.  The  only  other  curve 
which  has  more  than  one  conjugate  is  a  circular  helix,  for  since 
p  and  T  are  constant,  A/C  can  be  given  any  value  whatever ;  both 
the  given  helix  and  the  circular  helices  conjugate  to  it  are  traced 
on  circular  cylinders  with  the  same  axis. 

20.  Tangent  surface  of  a  curve.  For  the  further  discussion  of 
the  properties  of  curves  it  is  necessary  to  introduce  certain  curves 
and  surfaces  which  can  be  associated  with  them.  However,  in  con 
sidering  these  surfaces  we  limit  our  discussion  to  those  properties 
which  have  to  do  with  the  associated  curves,  and  leave  other  con 
siderations  to  their  proper  places  in  later  chapters. 

The  totality  of  all  the  points  on  the  tangents  to  a  twisted  curve  C 
constitute  the  tangent  surface  of  the  curve.  As  thus  defined,  the  sur 
face  consists  of  an  infinity  of  straight  lines,  which  are  called  the 
generators  of  the  surface.  Any  point  P  on  this  surface  lies  on  one 
of  these  lines,  and  is  determined  by  this  line  and  the  distance  t  from 
P  to  the  point  M  where  the  line  touches  the  curve,  as  is  shown  in 
fig.  7.  If  the  coordinates  x,  y,  z  of  M  are  expressed  in  terms  of  the 
arc  «,  the  coordinates  of  P  are  given  by 

(102)  f 


42  CURVES  IN  SPACE 

where  the  accents  denote  differentiation  with  respect  to  s.    When 
the  equations  of  the  curve  have  the  general  form 


the  coordinates  of  P  can  be  expressed  thus  : 

(103)    £=./»+«/»,    '?=/,(«)  +»./»,     f  -/,(«) 

where  v  =  -- 


From  this  it  is  seen  that  v  is  equal  to  the  distance  MP  only  when  s 
is  the  parameter. 

As  given  by  equations  (102)  or  (103),  the  coordinates  of  a  point 
on  the  tangent  surface  are  functions  of  two  parameters.  A  rela 

tion  between  these  parameters, 

such  as 

(104)          f(s,  t)  =  0, 

defines  a  curve  which  lies  upon 
the  surface.     For,  when  this 
FIG.  7  equation  is  solved  for  t  in  terms 

of  s  and  the  resulting  expres 

sion  is  substituted  in  (102),  the   coordinates   f,  ?;,  f  are 
functions  of  a  single  parameter,   and   consequently  the 
locus  of  the  point  (f,  77,  f)  is  a  curve  (§1). 

By  definition,  the  element  of  arc  of  this  curve  is  given  by 
da-2  =  di;  2  -f  drf  +  c?f  2.  This  is  expressible  by  means  of  (102)  and 
(41)  in  the  form 

(105)  d<rz  =    l  +  -2   ds2  +  2dsdt  +  dt\ 


where  t  is  supposed  to  be  the  expression  in  s  obtained  from  (104), 
and  p  is  the  radius  of  curvature  of  the  curve  (7,  of  which  the  sur 
face  is  the  tangent  surface.  This  result  is  true  whatever  be  the 
relation  (104).  Hence  equation  (105)  gives  the  element  of  length 
of  any  curve  on  the  surface,  and  do-  is  called  the  linear  element  of 
the  surface. 

According  as  t  in  equations  (102)  has  a  positive  or  negative 
value,  the  point  lies  on  the  portion  of  the  tangent  drawn  in  the 


TANGENT  SUBFACE  OF  A  CUKVE 


43 


positive  direction  from  the  curve  or  in  the  opposite  direction.  It 
is  now  our  purpose  to  get  an  idea  of  the  form  of  the  surface  in  the 
neighborhood  of  the  curve. 

In  consequence  of  (53)  equations  (102)  can  be  written 

1  \     L       1 


-^r+.-.u, 


6  pr 

The  plane  f  =  0  cuts  the  surface  in  a  curve  F.    The  point  MQ  of  (7, 

at  which  s  =  0,  is  also  a  point  of  F.    From  the  above  expression 

for  f  it  is  seen  that  for  points  of  F  near 

MQ  the  parameters  s  and  t  differ  only  in 

sign.    Hence,  neglecting  powers  of  s  and 

t  of  higher  orders,  the  equations  of  F  in 

the  neighborhood  of  J/0  are 

f=0,  ,=-£.,  r=_— <•• 

2/o  3  pr 

By  eliminating  t  from  the  last  two  equa 
tions,  we  find  that  in  the  neighborhood  of 
MQ  the  curve  F  has  the  form  of  a  semi- 
cubical  parabola  with  the  T^-axis,  that  is 
the  principal  normal  to  (7,  for  cuspidal 
tangent.  Since  any  point  of  the  curve  C 
can  be  taken  for  Jf0,  we  have  the  theorem : 

The  tangent  surface  of  a  curve  consists  of  two  sheets,  corresponding 
respectively  to  positive  and  negative  values  of  t,  which  are  tangent  to 
one  another  along  the  curve,  and  thus  form  a  sharp  edge. 

On  this  account  the  curve  is  called  the  edge  of  regression  of  the 
surface.  An  idea  of  the  form  of  the  surface  may  be  had  from  fig.  8. 

21.  Involutes  and  evolutes  of  a  curve.  When  the  tangents  of  a 
curve  C  are  normal  to  a  curve  Cv  the  latter  is  called  an  involute  of 
(7,  and  C  is  called  an  evolute  of  Cr  As  thus  defined,  the  involutes 
of  a  twisted  curve  lie  upon  its  tangent  surface,  and  those  of  a 


44 


CURVES  IN  SPACE 


plane  curve  in  its  plane.  The  latter  is  only  a  particular  case  of  the 
former,  so  that  the  problem  of  finding  the  involutes  of  a  curve  is 
that  of  finding  the  curves  upon  the  tangent  surface  which  cut  the 
generators  orthogonally. 

We  write  the  equations  of  the  tangent  surface  in  the  form 


Assuming  that  s  is  the  parameter  of  the  curve,  the  problem  reduces 
to  the  determination  of  a  relation  between  t  and  s  such  that 


By  means  of  (50)  this  reduces  to  dt  +  ds  =  0,  so  that  t  —  c  —  s, 
where  c  is  an  arbitrary  constant.  Hence  the  coordinates  x^  yv  zl 
of  an  involute  are  expressible  in  the  form 

(106)     2^=  x  +  a(c  —  s),     #!  =  #  +  £(<?—«),     zx=  z  +  ?(<?  —  s). 


Corresponding  to  each  value  of  c  there  is  an  involute ;  consequently 
a  curve  has  an  infinity  of  involutes.  If  two  involutes  correspond 
to  values  c^  and  c2  of  c,  the  segment  of  each  tangent  between  the 
curves  is  of  length  cl—  c2.  Hence  the  involutes  are  said  to  form  a 

system  of  parallel  curves  on  the 
tangent  surface. 

When  s  is  known  the  involutes 
are  given  directly  by  equations 
(106).  Hence  the  complete  de 
termination  of  the  involutes  of  a 
given  curve  requires  one  quad 
rature  at  most. 

From  the  definition  of  t  and 
its  above  value,  an  involute  can 
be  generated  mechanically  in  the  following  manner,  as  represented 
in  fig.  9.  Take  a  string  of  length  c  and  bring  it  into  coincidence 
with  the  curve,  with  one  end  at  the  point  s  =  0 ;  call  the  other 
end  A.  If  the  former  point  be  fixed  and  the  string  be  unwound 
gradually  from  the  curve  beginning  at  A,  this  point  will  trace  out 
an  involute  on  the  tangent  surface. 

By  differentiating  equations  (106),  we  get 

7         I  (c  —  s}  j         ,         m  (c  —  s)  j         ,        n  (c  —  s)  , 
dxl  = ds,     dyl  =  — ds,     dzl  —  — - ds. 


FIG.  9 


INVOLUTES  AND  EVOLUTES  45 

Hence  the  tangent  to  an  involute  is  parallel  to  the  principal  nor 
mal  of  the  curve  at  the  corresponding  point,  and  consequently  the 
tangents  at  these  points  are  perpendicular  to  one  another. 

As  an  example  of  the  foregoing  theory,  we  determine  the  involutes  of  the  cir 
cular  helix,  whose  equations  are 

x  =  a  cos  te,    y  =  a  sin  M,     z  =  au  cot  0, 

where  a  is  the  radius  of  the  cylinder  and  6  the  constant  angle  which  the  tangent  to 
the  curve  makes  with  axis  of  the  cylinder.  Now 

—  sin  u,  cos  M,  cot  0 

s  —  a  cosec  6  -  u,        a,  /3,  7  =  -  :  -- 

cosec  0 

Hence  the  equations  of  the  involutes  are 

Xi  =  a  cos  u  +  (au  —  c  sin  0)sin  M,     yi  =  a  sin  u  —  (aw  —  c  sin  0)  cos  u,     zi  =  c  cos  0. 

From  the  last  of  these  equations  it  follows  that  the  involutes  are  plane  curves 
whose  planes  are  normal  to  the  axis  of  the  cylinder,  and  from  the  expressions  for 
x\  and  yi  it  is  seen  that  these  curves  are  the  involutes  of  the  circular  sections  of 
the  cylinder. 

We  proceed  to  the  inverse  problem  : 
Given  a  curve  C,  to  find  its  evolutes. 

The  problem  reduces  to  the  determination  of  a  succession  of 
normals  to  C  which  are  tangent  to  a  curve  G'0.  If  MQ  be  the  point 
on  C0  corresponding  to  M  on  (7,  it  lies  in  the  normal  plane  to  C  at 
If,  and  consequently  its  coordinates  are  of  the  form 


where  p  and  q  are  the  distances  from  MQ  to  the  binomial  and  prin 
cipal  normal  respectively.  These  quantities  p  and  q  mi^t  be  such 
that  the  line  MQM  is  tangent  to  the  locus  of  Jf0  at  tiis  point,  that 
is,  we  must  have 


«  ,, 

where  /c  denotes  a  factor  of  proportionality.     "When  the  above 
values  are  substituted  in  these  equations,  we     et 


P 

and  two  other  equations  obtained  by  replacing  a,  Z,  X  by  /3,  m,  ft 
and  7,  n,  v.    Hence  the  expressions  in  parentheses  vanish.    From 


46  CURVES  IN  SPACE 

the  first  it  follows  that  p  is  equal  to  /o;  consequently  MQ  lies  on 
the  polar  line  of  C  at  M.    The  other  equations  of  condition  can  be 

written  ,  , 

dp      q  da      p 

J:  +  1  +  p«  «  a,        *  _  £  +       0> 

ds      r  ds      T 

Eliminating  /c,  we  get 


p 

For  the  sake  of  convenience  we   put  <o  =  I  -—  >  and  obtain  by 

integration 

-  =  tan  (o>  +  c), 
P 

where  c  is  the  constant  of  integration.  As  c  is  arbitrary,  there  is 
an  infinity  of  evolutes  of  the  curve  C\  they  are  defined  by  the 
following  equations,  in  which  c  is  constant  for  an  evolute  but 
changes  with  it: 

xQ=x+lp  +  \p  tan(o)  +  c),         yQ=y  +  mp  +  fip  tan(o>  -f  c), 

ZQ  =  z  -f  np  +  vp  tan  (o>  -f-  c). 

From  the  definition  of  q  it  follows  that  q/p  is  equal  to  the  tangent 
of  the  angle  which  MMQ  makes  with  the  principal  normal  to  C  at  M. 
Calling  this  angle  0,  we  have  6  =  &>  +  c.  The  foregoing  results  give 
the  following  theorem  : 

A  curve  C  admits  of  an  infinity  of  evolutes;  when  each  of  the 
normals  f"  (7,  which  are  tangent  to  one  of  its  evolutes,  is  turned 
through  the  sa^g  angle  in  the  corresponding  normal  plane  to  C,  these 
new  normals  are  tangent  to  another  evolute  of  C. 

In  fig.  5  the  locus  of  the  points  E  is  an  evolute  of  the  given 
curve. 

Each  system  o;  normals  to  C  which  are  tangent  to  an  evolute  C0 
constitute  a  tangent  surface  of  which  C0  is  the  edge  of  regression. 
Hence  the  evolves  of  C  are  the  edges  of  regression  of  an  infinity 
of  tangent  svffaces?  all  of  which  pass  through  C.. 

From  tbe  definition  of  w  it  follows  that  w  is  constant  only  when  the  curve  C 
is  plane.  jn  this  case  we  may  take  w  equal  to  zero.  Then  when  c  —  0  we  have 
the  evol\te  C0  in  the  plane  of  the  curve.  The  other  evolutes  lie  upon  the  right 


MINIMAL  CUEVES  47 

cylinder  formed  of  the  normals  to  the  plane  at  points  of  Co,  and  cut  the  elements 
of  the  cylinder  under  the  constant  angle  00°  —  c,  and  consequently  are  helices. 
Hence  we  have  the  theorem  : 

The  evolutes  of  a  plane  curve  are  the  helices  traced  on  the  right  cylinder  whose 
base  is  the  plane  evolute.  Conversely,  every  cylindrical  helix  is  the  evolute  of  an 
infinity  of  plane  curves. 

EXAMPLES 

1.  Find  the  coordinates  of  the  center  of  the  osculating  sphere  of  the  twisted 
cubic. 

2.  The  angle  between  the  radius  of  the  osculating  sphere  for  any  curve  and  the 
locus  of  the  center  of  the  sphere  is  equal  to  the  angle  between  the  radius  of  the 
osculating  circle  and  the  locus  of  the  center  of  curvature. 

3.  The  locus  of  the  center  of  curvature  of  a  curve  is  an  evolute  only  when  the 
curve  is  plane. 

4.  Find  the  radii  of  first  and  second  curvature  of  the  curve  x  =  a  sin  u  cos  w, 
y  =  a  cos2  it,  z  =  asinw.    Show  that  the  curve  is  spherical,  and  give  a  geometrical 
construction.    Find  its  evolutes. 

5.  Derive  the  properties  of  Bertrand  curves  (§  10)  without  the  use  of  the  moving 
trihedral. 

6.  Find  the  involutes  and  evolutes  of  the  twisted  cubic. 

7.  Determine  whether  there  is  a  curve  whose  bmormals  are  the  binormals  of  a 
second  curve. 

8.  Derive  the  results  of  §  21  by  means  of  the  moving  trihedral. 

22.  Minimal  curves.  In  the  preceding  discussion  we  have  made 
exception  of  the  curves,  defined  by 

z  =/!  (w),        y  =/a  (u),        z  =/8  (u), 
when  these  functions  satisfy  the  condition 


As  these  imaginary  curves  are  of  interest  in  certain  parts  of  the 
theory  of  surfaces,  we  devote  this  closing  section  to  their  discussion. 
The  equation  of  condition  may  be  written  in  the  form 


_  f         f  __  if 

J3  Jl  V2 

where  v  is  a  constant  or  a  function  of  u.    These  equations  are 
equivalent  to  the  following : 

(108)  ^:^lz*SI.<l+*>:.. 


48  CURVES  IN  SPACE 

At  most,  the  common  ratio  is  a  function  of  M,  say  f(u).  And  so 
if  we  disregard  additive  constants  of  integration,  as  they  can  be 
removed  by  a  translation  of  the  curve  in  space,  we  can  replace 
the  above  equations  by 


(1  09)       x  =.f(u)du,     y  =  if(n)du,     z  = 

We  consider  first  the  case  when  v  is  constant  and  call  it  a.  If 
we  change  the  parameter  of  the  curve  by  replacing  I  f(u)  du  by  a 
new  parameter  which  we  call  w,  we  have,  without  loss  of  generality, 

1-a2  .1+a2 

(110)  x  =  —^—ut          y  =  i—^--u,          z  =  au. 

For  each  value  of  a  these  are  the  equations  of  an  imaginary 
straight  line  through  the  origin.  Eliminating  #,  we  find  that  the 
envelope  of  these  lines  is  the  imaginary  cone,  with  vertex  at  the 
origin,  whose  equation  is 

(111)  z2+2/2+z2=0. 

Every  point  on  the  cone  is  at  zero  distance  from  the  vertex,  and 
from  the  equations  of  the  lines  it  is  seen  that  the  distance  between 
any  two  points  on  a  line  is  zero.  We  call  these  generators  of  the 
cone  minimal  straight  lines.  Through  any  point  in  space  there  are 
an  infinity  of  them  ;  their  direction-cosines  are  proportional  to 


—  '  —  »  > 

where  a  is  arbitrary.  The  locus  of  these  lines  is  the  cone  whose 
vertex  is  at  the  point  and  whose  generators  pass  'through  the  circle 
at  infinity.  For,  the  equation  in  homogeneous  coordinates  of  the 
sphere  of  unit  radius  and  center  at  the  origin  is  3?  +  y2  +  z2  =  w2, 
so  that  the  equations  of  the  circle  at  infinity  are 


Hence  the  cone  (111)  passes  through  the  circle  at  infinity. 

We  consider  now  the  case  where  v  in  equations  (109)  is  a  function 
of  u.  If  we  take  this  function  of  u  for  a  new  parameter,  and  for 
convenience  call  it  it,  equations  (109)  may  be  written  in  the  form 

(112)      g  _ll~p  £>(«)<**,    y  =  i£^^F(u)du,    z=  CuF(u)du, 


MINIMAL  CUEVES  49 

where,  as  is  seen  from  (108),  F(u)  can  be  any  function  of  u  different 
from  zero. 

If  we  replace  F(u)  by  the  third  derivative  of  a  function  f(u), 
thus  F(u)=f'"(u],  equations  (112)  can  be  integrated  by  parts  and  put 

in  the  form 

uf'(u)-f(u), 

(113)  1y-4 


Since  F  must  be  different  from  zero,  f(u)  can  have  any  form  other 
than  clu*+  c2u  -f  c8,  where  ^,  c2,  c3  are  arbitrary  constants. 


EXAMPLES 

1.  Show  that  the  tangents  to  a  minimal  curve  are  minimal  lines,  and  that  a 
curve  whose  tangents  are  minimal  lines  is  minimal. 

2.  Show  that  the  osculating  plane  of  a  minimal  curve  can  be  written  (X  —  x)  A 
+  (Y-y)B  +  (Z-z)C  =  Q,  where  A2  +  B2  +  C'2  =  0.    A  plane  whose  equation  is 
of  this  sort  is  called  an  isotropic  plane. 

3.  Show  that  through  each  point  of  a  plane  two  minimal  straight  lines  pass 
which  lie  in  the  latter. 

4.  Determine  the  order  of  the  minimal  curves  for  which  the  function /in  (113) 
satisfies  the  condition  4/'"/v  -  5/iv2  =  0. 

5.  Show  that  the  equations  of  a  minimal  curve,  for  which /in  (113)  satisfies  the 
condition  4/'"/v  —  5/iv2  =  a////3,  where  a  is  a  constant,  can  be  put  in  the  form 

8  8   .  8i, 

x  =  -  cos  £,     y  —  -  sin  £,     z  =  —  t. 


GENERAL  EXAMPLES 

1.  Show  that  the  equations  of  any  plane  curve  can  be  put  in  the  form 

x=J*cos0/(0)d0,         y  -  J  sin  0/(0)  d0, 

and  determine  the  geometrical  significance  of  0. 

2.  Prove  that  the  necessary  and  sufficient  condition  that  the  parameter  u  in  the 
equations  x  =fi(u),  y  =f2(u)  have  the  significance  of  0  in  Ex.  1  is 


3.  Prove  that  the  general  projective  transformation  transforms  an  osculating 
plane  of  a  curve  into  an  osculating  plane  of  the  transform. 

4.  The  principal  normal  to  a  curve  is  normal  to  the  locus  of  the  centers  of 
curvature  at  the  points  where  p  is  a  maximum  or  minimum. 


50  CURVES  IN  SPACE 

5  .  A  certain  plane  curve  possesses  the  property  that  if  C  be  its  center  of  curva 
ture  for  a  point  P,  Q  the  projection  of  P  on  the  x-axis,  and  T  the  point  where  the 
tangent  at  P  meets  this  axis,  the  area  of  the  triangle  CQT  is  constant.  Find  the 
equations  of  the  curve  in  terms  of  the  angle  which  the  tangent  forms  with  the  x-axis. 

6.  The  binormal  at  a  point  Mot  a  curve  is  the  limiting  position  of  the  common 
perpendicular  to  the  tangents  at  M  and  M',  as  N'  approaches  M. 

7.  The  tangents  to  the  spherical  indicatrices  of  the  tangent  and  binormal  of  a 
twisted  curve  at  corresponding  points  are  parallel. 

8.  Any  curve  upon  the  unit  sphere  serves  for  the  spherical  indicatrix  of  the 
binormal  of  a  curve  of  constant  torsion.    Find  the  coordinates  of  the  curve. 

9.  The  equations 

r  Idk  -  kdl  r  hdl  -  Idh  r  kdh  ~  hdk 

x  —  a  I  -  —  —  i      y  —  a  I  —  —  —  —  '      z  —  d  I 

J    #2  +  £2  +  12  J   hZ  +  fc2  +  1-2  J   h2  + 


where  a  is  constant  and  h,  k,  I  are  functions  of  a  single  parameter,  define  a  curve 
whose  radius  of  torsion  is  a. 


10.  If,  in  Ex.  9,  we  have 


k  =  sm/i0  +  -%sinX0      I  =  2    -    cos 


\/  2 

where  X  and  /x  are  constants  whose  ratio  is  commensurable,  the  integrands  are 
expressible  as  linear  homogeneous  functions  of  sines  and  cosines  of  multiples  of  0, 
and  consequently  the  curve  is  algebraic. 

11.  Equations  (1)  define  a  family  of  circles,  if  a,  &,  r  are  functions  of  a  parameter 
t.  Show  that  the  determination  of  their  orthogonal  trajectories  requires  the  solution 
of  the  Riccati  equation, 

*!  =  l*?,__L*(i-"), 

dt      r  dt        8rdr 

where  0=tanw/2. 

12.  Find  the  vector  representing  the  rate  of  change  of  the  acceleration  of  a 
moving  point. 

13.  When  a  curve  is  spherical,  the  center  of  curvature  for  the  point  is  the  foot 
of  the  perpendicular  upon  the  osculating  plane  at  the  point  from  the  center  of  the 
sphere. 

14.  The  radii  of  first  and  second  curvature  of  a  curve  which  lies  upon  a  sphere 
and  cuts  the  meridians  under  constant  angle  are  in  the  relation  1  +  ar  +  fy>2r  =  0, 
where  a  and  b  are  constants. 

15.  An  epitrochoidal  curve  is  generated  by  a  point  in  the  plane  of  a  circle  which 
rolls,  without  slipping,  on  another  circle,  whose  plane  meets  the  plane  of  the  first 
circle  under  constant  angle.  Find  its  equations  and  show  that  it  is  a  spherical  curve. 

16.  If  two  curves  are  in  a  one-to-one  correspondence  with  the  tangents  at 
corresponding  points  parallel,  the  principal  normals  at  these  points  are  parallel 
and  likewise  the  binormals  ;  two  curves  so  related  are  said  to  be  deducible  from 
one  another  by  a  transformation  of  Combescure. 

17.  If  two  curves  are  in  a  one-to-one  correspondence  and  the  osculating  planes 
at  corresponding  points  are  parallel,  either  curve  can  be  obtained  from  the  other 
by  a  transformation  of  Combescure. 


GENERAL  EXAMPLES  51 

18.  Show  that  the  radius   of   the  osculating  sphere  of  a  curve  is  given  by 
E'2  —  T2p4  [x""2  +  y"/z  +  z///2]  —  r2,  where  the  prime  denotes  differentiation  with 
respect  to  the  arc. 

19.  At  corresponding  points  of  a  twisted  curve  and  the  locus  of  the  center  of 
its  osculating  sphere  the  principal  normals  are  parallel,  and  the  tangent  to  one 
curve  is  parallel  to  the  binorinal  to  the  other ;  also  the  product?  of  the  radii  of 
torsion  of  the  two  curves  is  equal  to  the  product  of  the  radii  of  first  curvature, 
or  to  within  the  sign,  according  as  the  positive  directions  of  the  principal  normals 
are  the  same  or  different. 

20.  Determine  the  twisted  curves  which  are  such  that  the  centers  of  the  spheres 
osculating  the  curve  of  centers  of  the  osculating  spheres  of  the  given  curve  are  points 
of  the  latter. 

21.  Show  that  the  binormals  to  a  curve  do  not  constitute  the  tangent  surface 
of  another  curve. 

22.  Determine  the  directions  of  the  principal  normal  and  binormal  to  an  involute 
of  a  given  curve. 

23.  Show  that  the  equations 

x  =  a  C<f>  (u)  sin  u  du,      y  =  a  \  $  (u)  cos  u  du,      z  —  a  f  4>(u)\l/(u)du, 


where  0 (u)  —  (1 4-  ^2  4-  ^/2)-  (1 4-  ^2)   *  and  \f/  (u)  is  any  function  whatever,  define  a 
curve  of  constant  curvature. 

24.  Prove  that  when  ^  (u)  =  tan  w,  in  example  23,  the  curve  is  algebraic. 

25.  Prove  that  in  order  that  the  principal  normals  of  a  curve  be  the  binor- 
mals  of  another,  the  relation  a  I — h  — )  =  -  must  hold,  where  a  and  6  are  con 
stants.    Show  that  such  curves  are  defined  by  equations  of  example  23  when 

(1  4.  ^2  _|_  ^/2)3  _|_  (1  _|_  ^2)3(^"  _j_  ^,\2 
0  =  . 

(1  4.  i//2)^(l  4- 1//2  4- 1//2)^ 

26.  Let  \i,  /ii,  »*i  be  the  coordinates  of  a  point  on  the  unit  sphere  expressed  as 
functions  of  the  arc  <TI  of  the  curve.    Show  that  the  equations 


x  =  ek  I  \idffi  —  k  cot  w  /  (MI^I 

y  =  ek  j  mdai  —  k  cot  w  \  (v\\{  —  v{\\]  d<?i, 

z  =  ek  I  vida-i  —  k  cot  w  |  (\i/4 


where  k  and  w  are  constant,  e  =  ±  1,  and  the  primes  indicate  differentiation  with 
respect  to  o-i,  define  a  Bertrand  curve  for  which  p  and  T  satisfy  the  relation  (97)  ; 
show  also  that  X1?  /t1?  v\  are  the  direction-cosines  of  the  binormal  to  the  conjugate 
curve. 


CHAPTER  II 

CURVILINEAR  COORDINATES  ON  A  SURFACE*    ENVELOPES 

23.  Parametric  equations  of  a  surface.  In  the  preceding  chapter 
we  have  seen  that  the  coordinates  of  a  point  on  the  tangent  surface 
of  a  curve  are  expressible  in  the  form 


(1)     x 
where 


fl(u), 


?=/«(«*), 


are  the  equations  of  the  curve,  and  v  is  proportional  to  the  distance 
between  the  points  (f ,  77,  f ),  (x,  y,  z)  on  the  same  generator.  Since 
the  coordinates  of  the  surface  are  expressed  by  (1)  as  functions  of 

two  independent  parameters  w,  v, 
the  equations  of  the  surface  may 
be  written 


Consider  also  a  sphere  of  radius 
a  whose  center  is  at  the  origin  0 
(fig.  10).  If  v  denotes  the  angle, 
measured  in  the  positive  sense, 
which  the  plane  through  the  z-axis 
and  a  point  M  of  the  sphere  makes 
with  the  #z-plane,  and  u  denotes  the  angle  between  the  radius  OM 
and  the  positive  z-axis,  the  coordinates  of  M  may  be  written 


FIG.  10 


(3) 


x  =  a  sin  u  cos  v,         y  =  a  sin  u  sin  v,         z  =  a  cos  u. 


Here,  again,  the  coordinates  of  any  point  on  the  sphere  are  ex 
pressible  as  functions  of  two  parameter^,  and  the  equations  of  the 
sphere  are  of  the  form  (2)*. 


*  Notice  that  in  this  case  /^  is  a  function  of  u  alone. 


PARAMETRIC  EQUATIONS  OF  A  SURFACE  53 

In  the  two  preceding  cases  the  functions  fv  /2,  /3  have  par 
ticular  forms.  We  consider  the  general  case  where  /1?  /2,  /3  are 
any  functions  of  two  independent  parameters  w,  v,  analytic  for  all 
values  of  u  and  v,  or  at  least  for  values  within  a  certain  domain. 
The  locus  of  the  point  whose  coordinates  are  given  by  (2)  for  all 
values  of  u  and  v  in  the  domain  is  called  a  surface.  And  equa 
tions  (2)  are  called  parametric  equations  of  the  surface. 

It  is  to  be  understood  that  one  or  more  of  the  functions  /  may 
involve  a  single  parameter.  For  instance,  any  cylinder  may  be 
defined  by  equations  of  the  form 

x  =fi  M»      y  =/2  M»      z  =/a  (u^ v)- 

If  we  replace  u  and  v  in  (2)  by  independent  functions  of  two 
other  parameters  uv  vv  thus 

(4)  u  =  Fl  (uv  v,),         v  =  Fz  (ul9  vj, 
the  resulting  equations  may  be  written 

(5)  x  =  fa  (u^  VJ,          y  =  fa  K,  vj,          z  =  fa  (ut,  vj. 

If  particular  values  of  ^  and  vl  be  substituted  in  (4)  and  the  result 
ing  values  of  u  and  v  be  substituted  in  (2),  we  obtain  the  values 
of  #,  y,  z  given  by  (5),  when  u^  and  t^  have  been  given  the  par 
ticular  values.  Hence  equations  (2)  and  (5)  define  the  same  sur 
face,  provided  that  Fl  and  F2  are  of  such  a  form  that  fa,  fa,  fa 
satisfy  the  general  conditions  imposed  upon  the  F's.  Hence  the 
equations  of  a  surface  may  be  expressed  in  parametric  form  in 
the  number  of  ways  of  the  generality  of  two  arbitrary  functions. 

Suppose  the  first  two  of  equations  (2)  solved  for  u  and  v  in 
terms  of  x  and  y,  and  let  u  =  Ft(x,  y),  v  =  F2(x,  y)  be  a  set  of 
solutions.  When  these  equations  are  taken  as  equations  (4), 
equations  (5)  become 

x  =  x,         y  =  y,         z  =f(x,  y}, 

which  may  be  replaced  by  the  single  relation, 

(6)  2  =/(*,  y). 

If  there  is  only  one  set  of  solutions  of  the  first  two  of  equations  (2), 
equation  (6)  defines  the  surface  as  completely  as  (2).  If,  however, 
there  are  n  sets  of  solutions,  the  surface  would  be  defined  by  n 
equations,  z  =ft(x^  y). 


54        CURVILINEAR  COORDINATES  ON  A  SURFACE 

It  may  be  said  that  equation  (6)  is  obtained  from  equations  (2) 
by  eliminating  u  and  v.  This  is  a  particular  form  of  elimination, 
the  more  general  giving  an  implicit  relation  between  x,  y,  z,  as 

(7)  F(x,y,z)=0. 

If  we  have  a  locus  of  points  whose  coordinates  satisfy  a  relation 
of  the  form  (6),  it  is  a  surface  in  the  above  sense.  For,  if  we  take 
x  and  y  equal  to  any  analytic  functions  of  u  and  v,  namely  f^  and 
/2,  and  substitute  in  (6),  we  obtain  z  =/8(w,  v). 

In  like  manner  equation  (7)  may  be  solved  for  z,  and  one  or  more 
equations  of  the  form  (6)  obtained,  unless  z  does  not  appear  in  (7). 
In  the  latter  case  there  is  a  relation  between  x  and  y  alone,  so  that 
the  surface  is  a  cylinder  whose  elements  are  parallel  to  the  z-axis, 
and  its  parametric  equations  are  of  the  form 

x  =/i  W»      y  =/2  M»      2  =/3 (w,  v). 

Hence  a  surface  can  be  denned  analytically  by  equations  (2), 
(6),  or  (7).  Of  these  forms  the  last  is  the  oldest.  It  was  used 
exclusively  until  the  time  of  Monge,  who  proposed  the  form  (6); 
the  latter  has  the  advantage  that  many  of  the  equations,  which 
define  properties  of  the  surface,  are  simpler  in  form  than  when 
equation  (7)  is  used.  The  parametric  method  of  definition  is  due 
to  Gauss.  In  many  respects  it  is  superior  to  both  of  the  other 
methods.  It  will  be  used  almost  entirely  in  the  following 
treatment. 

24.  Parametric  curves.  When  the  parameter  u  in  equations  (2) 
is  put  equal  to  a  constant,  the  resulting  equations'  define  a  curve  on 
the  surface  for  which  v  is  the  parameter.  If  we  let  u  vary  continu 
ously,  we  get  a  continuous  array  of  curves  whose  totality  consti 
tutes  the  surface.  Hence  a  surface  may  be  considered  as  generated 
by  the  motion  of  a  curve.  Thus  the  tangent  surface  of  a  curve  is 
described  by  the  tangent  as  the  point  of  contact  moves  along  the 
curve ;  and  a  sphere  results  from  the  revolution  of  a  circle  about 
a  diameter. 

We  have  just  seen  that  upon  a  surface  (2)  there  lie  an  infinity 
of  curves  whose  equations  are  given  by  equations  (2),  when  u  is 
constant,  each  constant  value  of  u  determining  a  curve.  We  call 
them  the  curves  u  =  const,  on  the  surface.  In  a  similar  way, 


PARAMETRIC  CURVES  55 

there  is  an  infinite  family  of  curves  v  =  const.*  The  curves  of 
these  two  families  are  called  the  parametric  curves  for  the  given 
equations  of  the  surface,  and  u  and  v  are  the  curvilinear  coordinates 
of  a  point  upon  the  surface.f  We  say  that  the  positive  direction 
of  a  parametric  curve  is  that  in  which  the  parameter  increases. 
If  we  replace  v  in  equations  (2)  by  a  function  of  w,  say 

(8)  v  =  <t>(u), 

the  coordinates  #,  y,  z  are  functions  of  a  single  parameter  w,  and 
consequently  the  locus  of  the  point  (#,  y,  z)  is  a  curve.  Hence 
equation  (8)  defines  a  curve  on  the  surface  (2).  For  example, 
the  equation  v  =  au  defines  a  helix  on  the  cylinder 

x  =  a  cos  w,         y  —  o>  sin  u,         z  =  v. 
Frequently  equation  (8)  is  written  in  the  implicit  form, 

(9)  F(u,  v)  =  0. 

Conversely,  any  curve  upon  the  surface  is  defined  by  an  equation 
of  this  form.  For,  if  t  be  the  parameter  of  the  curve,  both  u  and  v  in 
equations  (2)  are  functions  of  t\  thus  w  =  ^1(Q,  v  =  (j>z(t).  Elimi 
nating  t  between  these  equations,  we  get  a  relation  such  as  (9). 

We  return  to  the  consideration  of  the  change  of  parameters, 
defined  by  equations  (4).  To  a  pair  of  values  of  u^  and  vl  there 
correspond  unique  values  of  u  and  v.  On  the  contrary,  it  may 
happen  that  another  pair  of  values  of  u^  and  vl  give  the  same 
values  of  u  and  v.  But  the  values  of  x,  y,  z  given  by  (5)  will  be 
the  same  in  both  cases  ;  this  follows  from  the  manner  in  which 
these  equations  were  derived.  On  this  account  when  equations  (4) 
are  solved  for  u^  and  vl  in  terms  of  u  and  v,  and  there  is  more 
than  one  set  of  solutions,  we  must  specify  which  solution  will 
be  used.  We  write  the  solution 

(10)  u^  =  <$>!  (w,  v),         v^  =  4>2  (u,  v). 

In  terms  of  the  original  parameters,  the  parametric  lines  u^=  const. 


and  vl  =  const,  have  the  equations, 


*  On  the  sphere  defined  by  equations  (3)  the  curves  v  —  const,  are  meridians  and 
u  —  const,  parallels. 

t  When  a  plane  is  referred  to  rectangular  coordinates,  the  parametric  lines  are  the 
two  families  of  straight  lines  parallel  to  the  coordinate  axes. 


56        CURVILINEAR  COORDINATES  ON  A  SURFACE 

where  a  and  b  denote  constants.  Unless  u  or  v  is  absent  from 
either  of  these  equations  the  curves  are  necessarily  distinct  from 
the  parametric  curves  u  =  const,  and  v  =  const.  Suppose,  now,  that 
v  does  not  appear  in  ^j  then  u^  is  constant  when  u  is  constant, 
and  vice  versa.  Consequently  a  curve  u^  =  const,  is  a  member  of 
the  family  of  curves  u  =  const.  Hence,  when  a  transformation  of 
parameters  is  made  by  means  of  equations  of  the  form 


or  ^=(2,,          ^(M), 

the  two  systems  of  parametric  curves  are  the  same,  the  difference 
being  in  the  value  of  the  parameter  which  is  constant  along  a  curve. 

EXAMPLES 

1  .  A  surface  which  is  the  locus  of  a  family  of  straight  lines,  which  meet  another 
straight  line  orthogonally  and  are  arranged  according  to  a  given  law,  is  called  a 
right  conoid  ;  its  equations  are  of  the  form  x  =  u  cos  v,  y  =  u  sin  u,  z  =  <j>  (v).  Show 
that  when  0  (v)  =  a  cot  v  +  b  the  conoid  is  a  hyperbolic  paraboloid. 

2.  Find  the  equations  of  the  right  conoid  whose  axis  is  the  axis  of  z,  and  which 

V2      z2 
passes  through  the  ellipse  x  —  a,  "—  -\  --  —  1. 

3.  When  a  sphere  of  radius  a  is  defined  by  (3),  find  the  relation  between  u  and 
v  along  the  curve  of  intersection  of  the  sphere  and  the  surface  x4  -f  y*  +  z4  =  £  a4. 
Show  that  the  curves  of  intersection  are  four  great  circles. 


4.  Upon  the  surface  x  —  v  w2  +  -J-  cos  t>,  y  —  Ma  4-  £  sin  v,  z  =  w,  determine  the 
curves  whose  tangents  make  with  the  z-axis  the  angle  tan-1  \/2.  Show  that  two 
of  these  curves  pass  through  every  point,  and  find  their  radii  of  first  and  second 
curvature. 

25.  Tangent  plane.  A  tangent  line  to  a  curve  upon  a  surface 
is  called  a  tangent  line  to  the  surface  at  the  point  of  contact.  It  is 
evident  that  there  are  an  infinity  of  tangent  lines  to  a  surface  at  a 
point.  We  shall  show  that  all  of  these  lines  lie  in  a  plane,  which 
is  called  the  tangent  plane  to  the  surface  at  the  point. 

To  this  end  we  consider  a  curve  C  upon  a  surface  and  let 
M(XJ  y,  z)  be  the  point  at  which  the  tangent  is  drawn.  The 
equations  of  the  tangent  are  (§  4) 

f-s  =  t)-y _ ?-g  =  ^ 

dx  dy  dz 

ds  ds  ds 


TANGENT  PLANE 


57 


where  f,  77,  f  are  the  coordinates  of  a  point  on  the  line,  depending 
for  their  values  upon  the  parameter  X.  If  the  equation  in  curvi 
linear  coordinates  of  the  curve  C  is  v  =  <f>(u),  the  above  equations 

may  be  written  .  *  \  j 

(  ^        ,  dx\  du 

',          dvl  ds 


^          ^^' 

=  \  —  -f-  4>  —  )-r 

\du          cv/  ds 


where  the  prime  indicates  differentiation.  In  order  to  obtain  the 
locus  of  these  tangent  lines,  we  eliminate  $'  arid  X  from  these 
equations.  This  gives 


(U) 


dx 

dy 

dz 

du 

du 

du 

dx 

dy 

dz 

dv 

~dv 

dv 

=  0, 


which  evidently  is  the  equation  of  a  plane  through  the  point  M. 
The  normal  to  this  plane  at  the  point  of  contact  is  called  the 
normal  to  the  surface  at  the  point. 

As  an  example,  we  find  the  equation  of  the  tangent  plane  to  the  tangent  surface 
of  a  curve  at  any  point.  If  the  values  from  (1)  be  substituted  in  equation  (11), 
the  resulting  equation  is  reducible  to 

(12)  /i  fi  fi 

fi  fi'          fs 

Hence  the  equation  of  the  tangent  plane  is  independent  of  u,  and  depends  only 
upon  u.  In  consequence  of  (I,  36)  *  we  have  the  theorem : 

The  tangent  plane  to  the  tangent  surface  of  a  curve  is  the  same  at  all  points  of  a 
generator;  it  is  the  osculating  plane  of  the  curve  at  the  point  where  the  generator 
touches  the  curve. 

When  the  surface  is  defined  by  an  equation  of  the  form  F(x,  y,  z)  =  0,  we 
imagine  that  x,  y.  z  are  functions  of  u  and  v,  and  differentiate  with  respect  to 
the  latter.  This  gives 

Hx  du      dy  du      dz  du  ~    '         dx  dv      dy  dv       dz  dv 

*  In  references  of  this  sort  the  Roman  numerals  refer  to  the  chapter. 


58         CURVILINEAR  COORDINATES  ON  A  SURFACE 

By  means  of  these  equations  the  equation  (11)  of  the  tangent  plane  can  be  given 
the  form 

(l-^+fo-jO^+tf-*)?^. 

ex  cy  cz 

When  the  Monge  form  of  the  equation  of  a  surface,  namely  z  =/(x,  y\t  is  used, 
it  is  customary  to  put 

dz  cz 

(14  r-  =  -P»         —  =  9- 

dx  cy 

Consequently  the  equation  of  the  tangent  plane  is 

(15)  (*  -  x)p  +  (77  -  y)q  -(f-z)  =  0. 

In  the  first  chapter  we  found  that  a  curve  is  defined  by  two  equations  of  the  form 

(16)  Fl(x,y,z)  =  0J  F2(x,  y,  z)  =  0. 

Hence  a  curve  is  the  locus  of  the  points  common  to  two  surfaces.    The  equa 
tions  of  the  tangent  to  the  curve  are 

g-X^q  -y  _£-Z^ 

dx          dy  dz 

where  cfcc,  dy,  dz  satisfy  the  relations 

5*1*  +  ^dy  +  ?*&  =  0,        ?*•*,  +  ^dy  +  ?*•*  =  0. 

dx  cy  cz  dx  cy  dz 

Consequently  the  equations  of  the  tangent  can  be  put  in  the  form 
£  -  z  77  -  y  $  -  z 


(17) 


dy  .  dz        dz     dy        dz    dx        dx     dz        dx    dy        dy     dx 


Comparing  this  result  with  (13),  we  see  that  the  tangent  line  to  a  curve  at  a  point  M 
is  the  intersection  of  the  tangent  planes  at  M  to  two  surfaces  which  intersect  along 
the  curve. 

*" 

EXAMPLES 

1.  Show  that  the  volume  of  the  tetrahedron  formed  by  the  coordinate  planes  and 
the  tangent  plane  at  any  point  of  the  surface  x  =  w,  y  =  u,  z  =  as/uv  is  constant. 

2.  Show  that  the  sum  of  the  squares  of  the  intercepts  of  the  axes  by  the  tan 
gent  plane  to  the  surface 

z  =  w3sin3u,        y  =  M3cos3v,        z  =  (a2  -  it2)*, 
at  any  point  is  constant. 

3.  Given  the  right  conoid  for  which  0(u)  =  a  sin  2  u.   Show  that  any  tangent 
plane  to  the  surface  cuts  it  in  an  ellipse,  and  that  if  perpendiculars  be  drawn 
to  the  generators  from  any  point  the  feet  of  the  perpendiculars  lie  in  a  plane 
ellipse. 


ENVELOPES  59 

4.  Show  that  the  tangent  planes,  at  points  of  a  generator,  to  the  right  conoid  for 
which  0  (u)  =  a  Vtan  u,  meet  the  plane  z  —  0  in  parallel  lines. 

5.  Find  the  equations  of  the  tangent  to  the  curve  whose  equations  are 

ax2  +  by*  +  cz2  =  1,         6x2  +  cy2  +  az2  =  1. 

6.  Find  the  equations  of  the  tangent  to  the  curve  whose  equations  are 

z(x  +  z)(x  —  a)  =  a3,        z(y  +  z)(y  —  a)  =  a3, 
and  show  that  the  curve  is  plane. 

7.  The  distance  from  a  point  M'  of  a  surface  to  the  tangent  plane  at  a  near-by 
point  M  is  of  the  second  order  when  MM  '  is  of  the  first  order  ;  and  for  other  planes 
through  M  the  distance  from  M  '  is  ordinarily  of  the  first  order. 

26.  One-parameter  families  of  surfaces.    Envelopes.   An  equation 
of  the  form 

(18)  F(x,  y,  z,a)  =  Q 

defines  an  infinity  of  surfaces,  each  surface  being  determined  by  a 
value  of  the  parameter  a.  Such  a  system  is  called  a  one-parameter 
family  of  surfaces.  For  example,  the  tangent  planes  to  the  tangent 
surface  of  a  twisted  curve  form  such  a  family. 

The  two  surfaces  corresponding  to  values  a  and  a'  of  the  param 
eter  meet  in  a  curve  whose  equations  may  be  written 

>  *  a) 


=  o. 


a  —  a 


As  a1  approaches  a,  this  curve  approaches  a  limiting  form  whose 
equations  are 

(19)  ^(W,«)=0,          »*(**«.«)-(). 

The  curve  thus  defined  is  called  the  characteristic  of  the  surface  of 
parameter  a.  As  a  varies  we  have  a  family  of  these  characteristics, 
and  their  locus,  called  the  envelope  of  the  family  of  surfaces,  is  a 
surface  whose  equation  is  obtained  by  eliminating  a  from  the  two 
equations  (19).  This  elimination  may  be  accomplished  by  solving 
the  second  of  (19)  for  a,  thus: 

a  =  $  (#,  y,  z), 
and  substituting  in  the  first  with  the  result 


60  ENVELOPES 

The  equation  of  the  tangent  plane  to  this  surface  is 


For  a  particular  value  of  a,  say  a0,  equations  (19)  define  the  curve 
in  which  the  surface  F(x,  y,  z,  a0)  =  0  meets  the  envelope  ;  and  from 
the  second  of  (19)  it  follows  that  at  all  points  of  this  curve  equa 
tion  (20)  of  the  tangent  plane  to  the  envelope  reduces  to 


This,  however,  is  the  equation  of  the  tangent  plane  to  the  surface 
F(x,  y,  z,  a0)  =  0.  If  we  say  that  two  surfaces  with  the  same  tan 
gent  plane  at  a  common  point  are  tangent  to  one  another,  we  have  : 

The  envelope  of  a  family  of  surfaces  of  one  parameter  is  tangent 
to  each  surface  along  the  characteristic  of  the  latter. 

The  equations  of  the  characteristic  of  the  surface  of  parameter  al  are 
(21) 

This  characteristic  meets  the  characteristic  (19)  in  the  point  whose 
coordinates  satisfy  (19)  and  (21),  or,  what  is  the  same  thing,  equa 
tions  (19)  and 

F(x,y,  z,  al)-F(x,^z,  a) 


As  al  approaches  0,  this  point  of  intersection  approaches  a  limiting 
position  whose  coordinates  satisfy  the  three  equations 

(22)  F-0,  ^=0,          !"     0. 

da  da2 

If  these  equations  be  solved  for  a;,  y,  z,  we  have 

(23)  *  =  /»,        y=/,(a),        *=/.(a). 

These  are  parametric  equations  of  a  curve,  which  is  called  the 
edge  of  regression  of  the  envelope. 


DEVELOPABLE  SURFACES 


61 


The  direction-cosines  of  the  tangent  to  the  edge  of  regression 

are  proportional  to  —  -»  -^,  —  -.    If  we  imagine  that  x,  y,  z  in  (19) 
da    da    da 

are  replaced  by  the  values  (23),  and  we  differentiate  these  equa 
tions  with  respect  to  a,  we  get,  in  consequence  of  (22), 


dx  da      dy  da 


dz  da 


tfF  dy 


*r  **  ,   -  -    -„  | 

da  d#  c?#      da  dy  da      da  dz  da 


dz 

—  ==  "• 


From  these  we  obtain 


dF 

dF 

dF 

dx 
da'' 

dy   dz 
da    da 

dx 

dy 

d2F 

~dz 
d2F 

dadx 

dady 

dadz 

But  from  (17)  it  follows  that  the  minors  of  the  right-hand  mem 
ber  are  proportional  to  the  direction-cosines  of  the  tangent  to  the 
curve  (19).  Hence  we  have  the  theorem: 

The  characteristics  of  a  family  of  surfaces  of  one  parameter  are 
tangent  to  the  edge  of  regression. 

27.  Developable  surfaces.  Rectifying  developable.  A  simple  ex 
ample  of  a  family  of  surfaces  of  one  parameter  is  afforded  by  a 
family  of  planes  of  one  parameter.  Their  envelope  is  called  a 
developable  surface ;  the  full  significance  of  this  term  will  be 
shown  later  (§43).  The  characteristics  are  straight  lines  which 
are  tangent  to  a  curve,  the  edge  of  regression.  When  the  edge  of 
regression  is  a  point,  the  surface  is  a  cone  or  cylinder,  according 
as  the  point  is  at  a  finite  or  infinite  distance.  We  exclude  this 
case  for  the  present  and  assume  that  the  coordinates  x,  y,  z  of  a 
point  on  the  edge  of  regression  are  expressed  in  terms  of  the  arc  s. 

We  may  write  the  equation  of  the  plane 


(24) 


(X-  x)a  +  (Y- 


where  «,  5,  c  also  are  functions  of  s.    The  characteristics  are  defined 
by  this  equation  and  its  derivative  with  respect  to  s,  namely : 

(25)        (X-  x)a'+(Y-  y)V+(Z-  z)c'-  ax'-  by'-  cz'=  0. 


62  ENVELOPES 

Since  these  equations  define  the  tangent  to  the  curve,  they  must 
be  equivalent  to  the  equations 

X-x  _Y-y  =Z—z 

x'  y'  z' 

Hence  we  must  have 

(26)  ax'  +  %'  +  czr  =  0,         a'x'  +  Vy'  +  c'z'  =  0. 

If  the  first  of  these  equations  be  differentiated  with  respect  to  s, 
the  resulting  equation  is  reducible,  in  consequence  of  the  second 

of  (26),  to  if  •  E  jf  ,      IT    A 

ax"  +  by"  +  cz"=  0. 

From  this  equation  and  (26)  we  find 

a  :  b  :  c  =  (y'z"-  z'y")  :  (z'x"-  x'z")  :  (x'y"-  y'x"}. 
Hence  by  (§  7)  we  have  the  theorem: 

On  the  envelope  of  a  one-parameter  family  of  planes  the  planes 
osculate  the  edge  of  regression. 

We  leave  it  to  the  reader  to  prove  that  the  edge  of  regres 
sion  of  the  osculating  planes  of  a  twisted  curve  is  the  curve 
itself. 

The  envelope  of  the  plane  normal  to  the  principal  normal  to 
a  curve  at  a  point  of  the  curve  is  called  the  rectifying  develop 
able  of  the  latter.  We  shall  find  the  equations  of  its  edge  of 
regression. 

The  equation  of  this  plane  is 

(27)  (X-  x)  I  +  (Y-  y)  m  +  (Z  -  z)  n  =  0. 

If  we  differentiate  this  equation  with  respect  to  the  arc  of  the  curve, 
and  make  use  of  the  Frenet  formulas  (I,  50),  we  obtain 

(28)  (I-       + 


From  these  equations  we  derive  the  equations  of  the  character 
istic  in  the  form 


RECTIFYING  DEVELOPABLE  63 

t  being  the  parameter  of  points  on  the  characteristic.  In  order  to 
find  the  value  of  t  corresponding  to  the  point  where  the  character 
istic  touches  the  edge  of  regression,  we  combine  these  equations 
with  the  derivative  of  (28)  with  respect  to  s,  namely  : 


and  obtain  (jL--J\t  +  -s*Q. 

VP       PT/       P 

Hence  the  coordinates  of  the  edge  of  regression  of  the  rectifying 
developable  are 

(29)      t  =  x 


, 

'p  —  p'r  TP  —  pr  T  p  —  pr 

Problem.  Under  what  conditions  does  the  equation  F(x,  y,  z)  =  0  define  a  devel 
opable  surface  ? 

We  assume  that  x,  y,  z  are  functions  of  two  parameters  w,  u,  such  that  the  curves 
u  =  const,  are  the  generators,  and  v  =  const,  are  any  other  lines.  The  equation  of 
the  tangent  plane  is 


This  equation  should  involve  u  and  be  independent  of  u.    Its  characteristic  is 
given  by  (i)  and 


where  we  have  put,  for  the  sake  of  brevity, 

.t.^.^w+y,  _ 

'  ' 


ax2  dxdy  dxdz      dx 


Since  equation  (i)  is  independent  of  u,  we  have 

(iii)  A*  +  B*  +  c£  =  0. 

dv          dv         at) 

Comparing  equations  (ii)  and  (iii)  with  (13),  we  see  that 

^-X^=0,        B-\^=0,        C-X^ 
3x  dy  oz 


64 


ENVELOPES 


where  X  denotes  a  factor  of  proportionality.   If  we  eliminate  X  —  x,  Y  —  y,  Z  —  z, 
and  X  from  these  equations  and  (i),  we  obtain  the  desired  condition 

2F  d2F  d^F  dF 
z2  dxdy  dxdz  dx 
2F  d2F  dzF  dF 


dx  dy  dy2 

dx  dz  dy  dz 

d_F_  aF 

dx  dy 


dy  dz  dy 

—  ?£ 

az2"  ~fa 

**  0 
dz 


=  0. 


EXAMPLES 

1.  Find  the  envelope  and  edge  of  regression  of  the  family  of  planes  normal  to 
a  given  curve. 

2.  Find  the  rectifying  developable  of  a  cylindrical  helix. 

3.  Prove  that  the  rectifying  developable  of  a  curve  is  the  polar  developable  of 
its  involutes,  and  conversely. 

4.  Find  the  edge  of  regression  of  the  envelope  of  the  planes 

x  sin  u  —  y  cos  u  -f  z  —  au  =  0. 

5.  Determine  the  envelope  of  a  one-parameter  family  of  planes  parallel  to  a 
given  line. 

6.  Given  a  one-parameter  family  of  planes  which  cut  the  xy-plane  under  con 
stant  angle ;   the  intersections  of  these  planes  with  the  latter  plane  envelop  a 
curve  C.    Show  that  the  edge  of  regression  of  the  envelope  of  the  planes  is  an 
evolute  of  C. 

7.  When  a  plane  curve  lies  on  a  developable  surface  its  plane  meets  the  tangent 
planes  to  the  surface  in  the  tangent  lines  to  the  curve.    Determine  the  developable 
surface  which  passes  through  a  parabola  and  the  circle,  described  in  a  perpendicular 
plane,  on  the  latus  rectum  for  diameter,  and  show  that  it  4s  a  cone. 

8.  Determine  the  developable  surface  which  passes  through  the  two  parabolas 
y2  =  4 ox,  z  =  0;  x2  =  4 ay,  z  =  6,  and  show  that  its  edge  of  regression  lies  on  the 
surface  y*z  =  x3(6  —  z). 

28.  Applications  of  the  moving  trihedral.  Problems  concern 
ing  the  envelope  of  a  family  of  surfaces  are  sometimes  more 
readily  solved  when  the  surfaces  are  referred  to  the  moving 
trihedral  of  a  curve,  which  is  associated  in  some  manner  with 
the  family  of  surfaces,  the  parameter  of  points  on  the  curve 
being  the  parameter  of  the  family. 

Let 
(30)  F(&  77,  £,  *)  =  0 


APPLICATIONS  OF  THE  MOVING  TRIHEDRAL        65 

define  such  a  family  of  surfaces.    Since  f,  77,  f  are  functions  of  *, 
the  equations  of  the  characteristics  are  (30)  and 

^-^^  +  ^^  +  ^^4.^=0 
ds  ~  d%  ds      dr)  ds       0f  ds       ds  ~ 

But  the  characteristics  being  fixed  in  space,  we  have  (I,  84) 


Hence  the  equations  of  the  characteristics  are 

(32)    ,_0,      /i 


If,  for  the  sake  of  brevity,  we  let  $(£,  ??,  f,  *)  =  0  denote  the  second 
of  these  equations,  the  edge  of  regression  is  defined  by  (32)  and 

<8S>      !( 

For  example,  the  family  of  osculating  planes  of  a  curve  is  defined  with  refer 
ence  to  the  moving  trihedral  by  f  =  0.  In  this  case  the  second  of  (32)  is  rj  =  0,  and 

(33)  is  -  4-  -  =  0.    Hence  the  tangents  are  the  characteristics,  and  the  edge  of  regres 

sion  is  the  curve  ;  for,  we  have  £  =  ??  =  f  =  0. 

In  like  manner  the  family  of  normal  planes  is  defined  by  £  =  0.  Now  the  second 
of  .(32)  is  17—  p=0  ;  consequently  the  polar  lines  are  the  characteristics.  Equation  (33) 
reduces  to  f  -f  p'r  =  0  ;  hence  the  locus  of  the  centers  of  the  osculating  spheres  is 
the  edge  of  regression  (cf.  §  18).  The  envelope  is  called  the  polar  developable. 

The  osculating  spheres  of  a  twisted  curve  constitute  a  family 
of  surfaces  which  is  readily  studied  by  the  foregoing  methods. 
From  (§  18)  it  follows  that  the  equation  of  these  spheres  is 


The  second  of  equations  (32)  for  this  case  is 


which,  since  spherical  curves  are  not  considered,  reduces  to  £=  0. 
And  equation  (33)  is  ??  =  0,  so  that  the  coordinates  of  the  edge  of 
regression  are  f  =  77  =  f  =  0.  Hence  : 

The  osculating  circles  of  a  curve  are  the  characteristics  of  its  oscu 
lating  spheres  ;  and  the  curve  itself  is  the  edge  of  regression  of  the 
envelope  of  the  spheres. 


66  ENVELOPES 

29.  Envelope  of  spheres.  Canal  surfaces.  We  consider  now  any 
family  of  spheres  of  one  parameter.  Referred  to  the  moving  tri 
hedral  of  the  curve  of  centers,  the  equation  of  the  spheres  is 


By  means  of  (32)  we  find  that  a  characteristic  is  the  circle  in 
which  a  sphere  is  cut  by  the  plane 


The  radius  of  this  circle  is  equal  to  rVl  —  rn.  Hence  the  char 
acteristic  is  imaginary  when  rn  >  1,  reduces  to  a  point  when 
r  =  ±  s  +  const.,  and  is  real  for  rf*  <  1. 

By  means  of  (33)  we  find  that  the  coordinates  of  the  edge  of 
regression  are  given  by 

(34)    f  =  -n-',     ,  =  [l-(rr')']p,     r 


Hence  the  edge  of  regression  consists  of  two  parts  with  corre 
sponding  points  symmetrically  placed  with  respect  to  the  oscu 
lating  plane  of  the  curve  of  centers  (7,  unless 


When  this  condition  is  satisfied  the  edge  is  a  single  curve,  and  its 
points  lie  in  the  osculating  planes  of  C.  We  have  seen  that  this 
is  the  case  with  the  osculating  spheres  of  a  curve.  We  shall  show 
that  when  the  above  condition  is  satisfied  the  spheres  osculate 
their  edge  of  regression  <7r 

We  write  the  above  equation  in  the  form     ^ 

(35)  p[l~-(rr')']  =  er^l-r'\ 

where  e  is  +  1  or  —  1,  so  that  p  may  be  positive. 

We  have  seen  (§  16)  that  the  absolute  and  relative  rates  of  change 
with  s  of  the  coordinates  f,  ?;,  f  of  a  point  on  Ct  are  in  the  relations 

M  =  ^_??  +  i,          ^  =  ^Z  +  l+f,         ^==^_!?. 
Ss      ds      p  &s      ds      p      T  Ss      ds      T 

When  the  values  (34)  are  substituted  in  the  right-hand  members 
of  these  equations,  we  obtain,  in  consequence  of  (35), 


ENVELOPE  OF  SPHEEES  67 

Hence  the  linear  element  Ss^  of  Cl  is  given  by 


cs1  = 
and 
(36)  Uo, 


Since  these  are  the  direction-cosines  of  the  tangent  to  C^  we  see 
that  this  tangent  is  normal  to  the  osculating  plane  to  the  curve  of 
centers  C.  Moreover,  these  direction-cosines  must  satisfy  (cf.  I,  83) 
the  equations 

/37\         8a  _da      b  8b  _  db      a      c  8c      do      b 

8s      ds      p  8s      ds      p      T  8s      ds      r 

Hence  we  have 


from  which  it  follows  that  the  radius  of  curvature  pl  of  Cl  is 
(38)  Pl=ee'rVT^, 

where  er  is  + 1  or  —  1 ,  so  that  /o1  may  be  positive.    Since,  now,  the 
direction-cosines  of  the  principal  normal  have  the  values 


it  follows  that  the  principal  normals  to  C  and  Cl  are  parallel. 
Furthermore,  since  these  quantities  must  satisfy  equations  (37), 
we  have  g3£  g3  ,  ^^  -. 

where  p[  denotes  the  derivative  of  pl  with  respect  to  sr  By  means 
of  (I,  51)  we  find  that  the  radius  of  torsion  rl  of  Cl  is  given  by 


.From  (38)  we  find  p[= — -»  so  that  the  radius  R^  of  the  oscu 
lating  sphere  of  C^  is  given  by  R*  —  p?  +  p[2 TI  =  r2,  and  consequently 
the  osculating  spheres  of  Cl  are  of  'the  same  radius  as  the  given 
spheres. 


68  ENVELOPES 

The  direction-cosines  of  the  tangent,  principal  normal,  and  binor 
mal  to  Cl  are  found  from  (36)  and  (39)  to  be 


Hence  the  coordinates  (I,  94)  of  the  center  of  the  osculating  sphere 
of  Cl  are  reducible,  in  consequence  of  (34),  to 

£  +  liPi  -  P(TI\  =  °'     *?  +  miPi  ~  friPi  =  °>      ?  +  W1p1  -  XT^  =  0. 
Therefore  we  have  the  theorem  : 

When  the  edge  of  regression  of  a  family  of  spheres  of  one  param 
eter  has  only  one  branch,  the  spheres  osculate  the  edge. 

Since  r  does  not  appear  in  equation  (35),  it  follows  that  when 
r  is  given  as  a  function  of  s,  the  intrinsic  equations  of  the  curve  of 
are 


where  the  function/(s)  is  arbitrary.  Moreover,  any  curve  will  serve 
for  the  curve  of  centers  of  such  an  envelope  of  spheres.  The  deter 
mination  of  r  requires  the  solution  of  equation  (35)  and  consequently 
involves  two  arbitrary  constants. 

When  all  the  spheres  of  a  family  have  the  same  radius,  the 
envelope  is  called  a  canal  surface.  From  (34)  it  is  seen  that  in 
this  case  a  characteristic  is  a  great  circle.  Moreover,  equation  (35) 
reduces  to  p  =  r.  Hence  a  necessary  and  sufficient  condition  that 
the  edge  of  regression  of  a  canal  surface  consist  of  a  single  curve 
is  that  the  curve  of  centers  be  of  constant  curvature  and  the  radius 
of  the  sphere  equal  to  the  radius  of  first  curvature  of  the  curve. 

GENERAL  EXAMPLES 

1.  Let  MN  be  a  generator  of  the  right  conoid 

x  =  u  cos  u,        y  =  u  sin  i>,         z  =  2  k  cosec  2  D, 

M  being  the  point  in  which  it  meets  the  z-axis.  Show  that  the  tangent  plane  at  N 
meets  the  surface  in  a  hyperbola  which  passes  through  M,  and  that  as  N  moves 
along  the  generator  the  tangent  at  M  to  the  hyperbola  describes  a  plane. 

2.  A  point  moves  on  an  ellipsoid  --  h  —  H  --  =  1,  so  that  the  direction  of  its 

a2      ft2       c2 

motion  always  passes  through  the  perpendicular  from  the  center  on  the  tangent 
plane  at  the  point.  Show  that  the  path  of  the  point  is  the  curve  in  which  the  ellipsoid 

is  cut  by  the  surface  xlymzn  =  const.  ,  where  1:  m  :  n  —  ---  :  --  —  :  ---  • 


GENERAL  EXAMPLES  69 

3.  If  each  of  the  generators  of  a  developable  surface  be  revolved  through  the 
same  angle  about  the  tangent  to  an  orthogonal  trajectory  of  the  generators  at  the 
point  of  intersection,  the  locus  of  these  lines  is  a  developable  surface  whose  edge 
of  regression  is  an  evolute  of  the  given  trajectory. 

4.  Show  that  the  edge  of  regression  of  the  family  of  planes 

(1  -  w2)z  +  i(l  +  u*)y  +  2uz  +f(u)  =  0 
is  a  minimal  curve. 

5.  The  developable  surface  which  passes  through  the  circles  x2  -f  y*  =  a2,  z  =  0; 
x2  -f  z2  —  62,  y  =  0  meets  the  plane  x  =  0  in  an  equilateral  hyperbola. 

6.  Find  the  edge  of  regression  of  the  developable  surface  which  envelopes  the 
surface  az  =  xy  along  the  curve  in  which  the  latter  is  cut  by  the  cylinder  x2  =  by. 

7.  Find  the  envelope  of  the  planes  which  pass  through  the  center  of  an  ellipsoid 
and  cut  it  in  sections  of  equal  area. 

8.  The  first  and  second  curvatures  of  the  edge  of  regression  of  the  family  of 
planes  ax  +  /3y  +  yz  —  p,  where  <r,  /3,  7,  p  are  functions  of  a  single  parameter 
u  and  a2  +  /32  -f  y2  =  1,  are  given  by 

1  A3  A2 


where 


A  = 


a    a'    a" 


p  p  p  p 

Of     off  -r\        oc.  a'  a"  a'" 

P     P      P      ,        D=    ft  0,  p,  p,, 

y  Y  y"  y'" 


9.  Derive  the  equations  of  the  edge  of  regression  of  the  rectifying  developable 
by  the  method  of  §  28. 

10.  Derive  the  results  of  §  29  without  the  aid  of  the  moving  trihedral. 

1 1 .  Find  the  envelope  of  the  spheres  whose  diameters  are  the  chords  of  a  circle 
through  a  point  of  the  latter. 

12.  Find  the  envelope  and  edge  of  regression  of  the  spheres  which  pass  through 
a  fixed  point  and  whose  centers  lie  on  a  given  curve. 

13.  Find  the  envelope  and  edge  of  regression  of  the  spheres  which  have  for 
diametral  planes  one  family  of  circular  sections  of  an  ellipsoid. 

14.  Find   the   envelope   and   edge   of   regression   of   the   family  of   ellipsoids 

(3^2         <j/2\          -j2 
1 H — -  =  1,  where  a  is  the  parameter. 

15.  Find  the  envelope  of  the  family  of  spheres  whose  diameters  are  parallel 
chords  of  an  ellipse. 

16.  Find  the  equations  of  the  canal  surface  whose  curve  of  centers  is  a  circular 
helix  and  whose  edge  of  regression  has  one  branch.    Determine  the  latter. 

17.  Find  the  envelope  of  the  family  of  cones 

(ax  +  x  +  y  +  z  -  1)  (ay  +  z)  -  ax  (x  +  y  +  z  -  1)  =  0, 
where  a  is  the  parameter. 

' 


CHAPTER  III 

LINEAR  ELEMENT  OF  A  SURFACE.    DIFFERENTIAL  PARAMETERS. 
CONFORMAL  REPRESENTATION 

30.  Linear  element.    Upon  a  surface  £,  defined  by  equations  in 
the  parametric  form 

(1)  x  =fi  (i*,  v),         y  =/a  (w,  v),          2  =/,  (M,  v), 

we  select  any  curve  and  write  its  equations  $  (u,  v)  —  0.    From  §  3 
we  have  that  the  linear  element  of  the  curve  is  given  by 

(2)  d? 


fa  j     t  foj       j        dij  ,     t  dy  ,       ,        dz  ,     ,  dz 

where   ax  =  —  du  -\  --  av,    a  y  =  —  au  +  -^  »vf    «2  =  —  aw  H  --- 

^M  ^v  ^w  9»  du  dv 

the  differentials  <^w,  dv  satisfying  the  condition 

2$  ,      a^  , 

—  c?w  +  —  dv  =  0. 

^w  dv 

ifweput 


du  dv      cu  cv      du  dv 


or,  in  abbreviated  form, 


equation  (2)  becomes 

(4)  oV2  =  Edu?  +  2  Fdudv  +  G  dv2. 

The  functions  E,  F,  G  thus  denned  were  first  used  by  Gauss.* 
When  the  surface  is  real,  and  likewise  the  curvilinear  coordinates 

*  Disquisitiones  generates  circa  superficies  curvas  (English  translation  by  Morehead 
and  Hiltebeitel),  p.  18.  Princeton,  1902.  Unless  otherwise  stated,  all  references  to  Gauss 
are  to  this  translation. 

70 


LINEAK  ELEMENT  71 


w,  v,  the  functions  Vj£,  #  are  real.  We  shall  understand  also  that 
the  latter  are  positive.  There  is,  however,  an  important  excep 
tional  case,  namely  when  both  E  and  G  are  zero  (cf.  §  35). 

For  any  other  curve  equation  (4)  will  have  the  same  form,  but 
the  relation  between  du  and  dv  will  depend  upon  the  curve. 
Consequently  the  value  of  c7s,  given  by  (4),  is  the  element  of  arc 
of  any  curve  upon  the  surface.  It  is  called  the  linear  element  of 
the  surface  (cf.  §  20).  However,  in  order  to  avoid  circumlocution, 
we  shall  frequently  call  the  expression  for  ds2  the  linear  element, 
that  is,  the  right-hand  member  of  equation  (4),  which  is  also  called 
the  first  fundamental  quadratic  form.  The  coefficients  of  the  lat 
ter,  namely  E,  F,  G,  are  called  the  fundamental  quantities  of  the 
first  order. 

If,  for  the  sake  of  brevity,  we  put 


(5) 


d(u,  v) 


du  du 

dz 


d(u,  v)  d(u,  v) 


dv   dv 

it  follows  from  (3)  and  (5)  that 
(6)  EG  -  F2=  A2  +  B2  +  C2. 

Hence  when  the  surface  is  real  and  likewise  the  parameters, 
the  quantity  EG—F2  is  different  from  zero  unless  J,  B,  and  C 
are  zero.  But  if  A,  B,  and  C  are  zero,  it  follows  from  (5)  that  u 
and  v  are  not  independent,  and  consequently  equations  (1)  define 
a  curve  and  not  a  surface.  However,  it  may  happen  that  for 
certain  values  of  u  and  v  all  the  quantities  J,  B,  C  vanish. 
The  corresponding  points  are  called  singular  points  of  the  sur 
face.  These  points  may  be  isolated  or  constitute  one  or  more 
curves  upon  the  surface ;  such  curves  are  called  singular  lines. 
In  the  following  discussion  only  ordinary  points  will  be  con 
sidered. 

From  the  preceding  remarks  it  follows  that  for  real  surfaces, 
referred  to  real  coordinate  lines,  the  function  H  defined  by 

(?)  1 


is  real,  and  it  is  positive  by  hypothesis. 


72  LINEAR  ELEMENT  OF  A  SURFACE 

31.  Isotropic  developable.  The  exceptional  case,  where  the  surface  is  imaginary 
and  H  is  zero,  is  afforded  by  the  tangent  surface  of  a  minimal  curve.  The  equa 
tions  of  such  a  surface  are  (cf  .  §  22) 

/I  —  w2  1  —  w2 

—  £—  0  (u)  du  +  —  —  0  (M)  u, 


=  ju(f>(u)  du  +  u<j>  (u)  v, 


where  0(w)  is  a  function  of  u  different  from  zero.  It  is  readily  found  that  J£=v202(w), 
F  =  G  —  0,  and  consequently  EG  —  F2  =  0.  This  equation  is  likewise  the  sufficient 
condition  that  the  surface  be  of  the  kind  sought.  For,  when  itjs  satisfied,  the  equa 
tion  of  the  linear  element  can  be_written  ds'2  =  (Vj£dw  +  V(?dw)2.  If  X  denote  an 
integrating  factor  of  \^Edu  -f-  V(?du,  and  a  function  MI  be  defined  by  the  equation 
\(\fEdu  +  VGdv)  —  dwi,  the  above  equation  becomes  ds2  =  —  duf.  Hence,  if  we 

A"** 

take  for  parametric  curves  u\  —  const,  and  any  other  system  for  vi  =  const.  ,  we 
have  FI  =  0,  GI  =  0.  In  other  form  these  equations  are 


3v 
In  accordance  with  the  last  equation  we  put 


01?!  2  8i?i 

where  A;  is  undetermined. 
By  integration  we  have 


r 


X,  M,  v  being  functions  of  HI  alone.    When  these  values  are  substituted  in  the  first 
of  the  above  equations  of  condition,  we  get 


to  be  satisfied  by  X,  /A,  and  v. 

The  equation  of  the  tangent  plane  to  the  surface  (i)  is  reducible  to 

(1  -  W]l2)  (X  -  x)  +  i  (1  +  w-2)  (  Y  -  y)  +  2  Ml  (Z  -  z)  =  0. 

Hence  the  surface  is  developable.    Since  its  edge  of  regression  is  a  minimal  curve 
(Ex.  4,  p.  69),  the  theorem  is  proved.    The  surface  is  called  an  isotropic  developable. 

32.  Transformation  of  coordinates.  It  is  readily  found  that  the 
functions  E,  F,  G  are  unaltered  in  value  by  any  change  of  the 
rectangular  axes.  But  now  we  shall  show  that  these  functions 
change  their  values  when  there  is  a  change  of  the  curvilinear 
coordinates. 


TRANSFORMATION  OF  COORDINATES 


73 


Let  the  transformation  of  coordinates  be  defined  by  the  equations 
(8)  u  =  u(u^  Vj),          v  =  v(ul,vl); 

then  we  have 

dx  _  dx  du      dx  dv  dx  _  dx  du      dx  dv 

du,      du  du,      dv  du.  dv.      du  dv,      dv  dv 

i  i  i  i  11 


we  find  the  relations 


(9) 


E—  — 

du  dv 


fa 

dv       du  dv 
dv.      dv,  du, 

1  11 


—  — 
d^  dvl 


Hence  the  fundamental  quantities  of  the  first  order  assume  new 
forms  when  there  is  a  change  of  curvilinear  coordinates. 
From  (8)  we  have,  by  differentiation, 

du             du  dv  dv 

du  =  —  dul  -\ aVj,          dv  —  —  du.  +  - —  dv.. 

du.  dv.  du,  dv, 

11  11 

Solving  these  equations  for  duv  dv^  we  get 

l/dv   ,        du  ,  \  1  /      dv    ,      .    du 


where 
(10) 

Hence  we  have 


d(u,  v) 


du      8  dvl        dv 


(du 


\  "du 


so  that 

(12)  1  ^ 

From  (9)  we  find  the  relation 


LINEAR  ELEMENT  OF  A  SURFACE 


By  means  of  this  equation  and  the  relations  (11),  we  can  transform 
equations  (9)  into  the  following : 


(13) 


^^\^2F—^—l  +  G(  — 
1  du 


EG— I* 

Efa1to1__F(faito1 


<  i- 


0V, 

du 


cu^  ct\ 
cu  cu 


•r^1 


EG- 


cv 


CM 


,T    /"«  Jj<2  Tfi  S~1  WA 

1^1 —      1  JG/Cr — .T 

33.  Angles  between  curves.  The  element  of  area.  Upon  a  para 
metric  line  v  =  const,  we  take  for  positive  sense  the  direction 
in  which  the  parameter  u  increases,  and  likewise  upon  a  curve 
u  =  const,  the  direction  in  which  v  increases.  If  efe,,  and  e?sM  denote 
the  elements  of  arc  of  curves  v  —  const,  and  u  =  const,  respec 
tively,  we  find,  from  (4), 

(14)  dsv  =  ^Edu,          dsu  =  ^Gdv. 

Hence,  if  aw,  #„,  yv  and  «M,  /9M,  yu  denote  the  direction-cosines  of  the 
tangents  to  these  curves  respectively,  we  have 


fa 

du 

1     dx 


cu  "*</ E  du 

cy  <  1    dz 


We  have  seen  that  through  an  ordinary  point  of  a  surface 
there  passes  one  curve  of  parameter  u  and  one  of  parameter  v. 
If,  as  in  fig.  11,  &)  denotes  the  angle,  between  0°  and  180°, 
formed  by  the  positive  directions  of  the  tangents  to  these  curves 
at  the  point,  we  have 


(15) 
and 
(16) 


cos  ft)  =  aa  + 


,  +  7«7W  =  -7== 


sin  &)  = 


VJSQ--F*      H 


ANGLES  BETWEEN  CUKVES 


75 


When  two  families  of  curves  upon  a  surface  are  such  that 
through  any  point  a  curve  of  each  family,  and  but  one,  passes, 
and  when,  moreover,  the  tangents  at  a  point  to  the  two  curves 
through  it  are  perpendicular,  the  curves  are  said  to  form  an 
orthogonal  system.  From  (15)  we  have  the  theorem: 

A  necessary  and  sufficient  condition  that  the  parametric  lines  upon 
a  surface  form  an  orthogonal  system  is  that  F  =  0. 

Consider  the  small  quadrilateral  (fig.  11)  whose  vertices  are 
the  points  with  the  curvilinear  coordinates  (u,  v),  (u  -f  du,  v), 
(u,  v  +  dv),  (u  +  du,  v  +  dv).  To  within  terms  of  higher  order  the 
opposite  sides  of  the  figure  are  equal.  Consequently  it  is  approxi 
mately  a  parallelogram  whose  sides 
are  of  length  v  E  du  and  \  G  dv  and 
the  included  angle  is  o>.  The  area 
of  this  parallelogram  is  called  the 
element  of  area  of  the  surface.  Its 
expression  is 

(17)   d^  =  sin  CD  V 'EG  dudv  =  H  dudv. 


/(u+du.v) 
FIG.  11 


If  C  is  any  curve  on  a  surface,  the  direction-cosines  a, 
its  tangent  at  a  point  have  the  form 

0      dy      /dy  du      dy  dv 

/-/    __  _.    __    ^     ,-^--    I     ^4^n    ,  _    __1  g<      _ 

ds      \cu  ds      dv  ds 


7  of 


dx   /dx  du   dx  dv 

_  _  I  _  _  _  _  I   _  _  _  _ 

ds   cu  ds   dv  ds 


dz  _  /dz  du      cz  dv 
J~'ds~\^uds      dv  ~ds 

If  we  put  dv/du  =  X  and  replace  ds  by  the  positive  square  root  of 
the  right-hand  member  of  (4),  the  above  expressions  can  be  written 


dv 


(18) 


du 


dv 


7  = 


du 


dv 


76  LINEAR  ELEMENT  OF  A  SURFACE 

From  these  results  it  is  seen  that  the  direction-cosines  depend 
not  upon  the  absolute  values  of  du  and  dv,  but  upon  their 
ratio  X.  The  value  of  X  is  obtained  by  differentiation  from  the 
equation  of  (7,  namely 

(19) 


Let  Cl  be  a  second  curve  meeting  C  at  a  point  M,  and  let 
the  direction-cosines  of  the  tangent  to  Cl  at  M  be  av  /3V  yr 
They  are  given  by 


"l     du  8s  '   dvW 

and  similar  expressions  for  /3t  and  7^  where  8  indicates  variation 
in  the  direction  of  Cr 

If  6  denotes  the  angle  between  the  positive  directions  to  C  and 
Cl  at  M,  we  have,  from  (18)  and  (20), 

Edu§u  +  F(du 8v  -f  dv  8u]  +Gdv8v 
(21)    cos  0  =  #tf x  +  ppj  +  77j  = i j-^ 

and 


sin  9  =  ±  Vl  -  cos'0  =  ±  H 


(8u  dv      8v  du 
8s  ds      8s  ds 


This  ambiguity  of  sign  is  due  to  the  fact  that  6  as  denned  is  one 
of  two  angles  which  together  are  equal  to  360°.  We  take  the 
upper  sign,  thus  determining  6.  This  gives 

/nft.  .    Q      Tr/8udv      8v  du\ 

(22)  sm6  =  H  -_  —  -__. 

\09  d8       08  ds/ 

The  significance  of  the  above  choice  will  be  pointed  out  shortly. 

When  in  particular  Cl  is  the  curve  v  =  const,  through  M,  we 
have  8v  =  0  and  8s  =  V ' E  8u,  so  that 

/00.  ~         1     { -^du      T,dv\  .    a        If  dv 

(23)  cos<90  =  — =    j£— +  .F— U         81X100=-=— 

ds          ds/ 


From  these  equations  we  obtain 
(24)  tan*0= 


Edu+Fdv 

The  angle  co  between  the  positive  half  tangents  to  the  para 
metric    curves   has    been    uniquely  denned.    Hence    there  is,  in 


ANGLES  BETWEEN  CURVES  77 

general,  only  one  sense  in  which  the  tangent  to  the  curve  v  =  const, 
can  be  brought  into  coincidence  with  the  tangent  to  the  curve 
u  =  const,  by  a  rotation  of  amount  co.  We  say  that  rotations  in 
this  direction  are  positive,  in  the  opposite  sense  negative.  From 
(23)  it  is  seen  that  00  is  the  angle  described  in  the  positive  sense 
when  the  positive  half  tangent  to  the  curve  v  =  const,  is  rotated 
into  coincidence  with  the  half  tangent  to  C.  And  so  in  the  general 
case  6,  defined  by  (22),  is  the  angle  described  in  the  positive  rota 
tion  from  the  second  curve  to  the  first. 

From  equations  (15),  (16),  and  (23)  we  find 

(26) 


These  equations  follow  also  directly  from  (20)  and  (21)  by  consid 
ering  the  curve  u  =  const,  as  the  second  line. 

As  an  immediate  consequence  of  equation  (21)  we  have  the 
theorem : 

A  necessary  and  sufficient  condition  that  the  tangents  to  two  curves 
upon  a  surface  at  a  point  of  meeting  be  perpendicular  is 

(26)  E  du  Su  +  F(du  Sv  +  dv  8u)  +  G  dv  $v  =  0. 

EXAMPLES 

1.  Show  that  when  the  equation  of  a  surface  is  of  the  form  z  =/(£,  y),  its  linear 
element  can  be  written 

ds2  =  (1  +  p2)  dx2  +  2pqdxdy  +  (l  +  q2)  dy2, 

where  p  =  dz/dx,  and  q  =  cz/dy.  Under  what  conditions  do  the  lines  x  =  const. , 
y  =  const,  form  an  orthogonal  system  ? 

2.  Show  that  the  parametric  curves  on  the  sphere 

x  =  a  sin  u  cos  v,        y  =  a  sin  u  sin  u,        z  =  a  cos  u 

form  an  orthogonal  system.  Determine  the  two  families  of  curves  which  meet  the 
curves  v  =  const,  under  the  angles  ir/4  and  3  7r/4.  Find  the  linear  element  of  the 
surface  when  these  new  curves  are  parametric. 

3.  Find  the  equation  of  a  curve  on  the  paraboloid  of  revolution  x  =  wcosu, 
y  =  itsinu,  z  =  w2/2,  which  meets  the  curves  v  =  const,  under  constant  angle  a 
and  passes  through  two  points  (MO,  i>o),  (MI,  «i).    Determine  a  as  a  function  of 

4.  Find  the  differential  equation  of  the  curves  upon  the  tangent  surface  of  a 
curve  which  cut  the  generators  under  constant  angle  a. 


78  LINEAR  ELEMENT  OF  A  SURFACE      . 

5.  Show  that  the  equations  of  a  curve  which  lies  upon  a  right  cone  and  cuts  all 
the  generators  under  the  same  angle  are  of  the  form  x  =  ce'"'  cosu,  y  —  ce°«sinw, 
2  =  6e°",  where  a,  6,  and  c  are  constants.    What  is  the  projection  of  the  curve  upon 
a  plane  perpendicular  to  the  axis  of  the  cone  ?   Find  the  radius  of  curvature  of 
the  curve. 

6.  Find  the  equations  of  the  curves  which  bisect  the  angles  between  the  para 
metric  curves  of  the  paraboloid  in  Ex.  3. 

34.  Families  of  curves.    An  equation  of  the  form 

(27)  <£(w,  v)=c, 

where  c  is  an  arbitrary  constant,  defines  an  infinity  of  curves,  or  a 
family  of  curves,  upon  the  surface.  Through  any  point  of  the  sur 
face  there  passes  a  curve  of  the  family.  For,  given  the  curvilinear 
coordinates  of  a  point,  when  these  values  are  substituted  in  (27) 
we  obtain  a  value  of  c,  say  CQ  ;  then  evidently  the  curve  $  =  c0 
passes  through  the  point.  We  inquire  whether  this  family  of 
curves  can  be  defined  by  another  equation.  Suppose  it  is  possible, 
and  let  the  equation  be 

(28)  ^(U,V)  =  K. 

Since  c  and  K  are  constant  along  any  curve  and  vary  in  passing 
from  one  curve  to  another,  each  is  necessarily  a  function  of  the 
other.  Hence  i|r  is  a  function  of  fa  Moreover,  if  ifr  is  any 
function  of  fa  equations  (27)  and  (28)  define  the  same  family  of 
curves. 

From  equation  (24)  it  is  seen  that  the  direction,  at  any  point,  of 
the  curve  of  the  family  through  the  point  is  determined  by  the 
value  of  dv/du.  We  obtain  the  latter  from  the  equation 

36  ,        d<f>  _ 

(29)  g*H.2*.i.<* 

which  is  derived  from  (27)  by  differentiation. 

Let  $  (u,  v)  =  cbe  an  integral  of  an  ordinary  differential  equation 
of  the  first  order  and  first  degree,  such  as 

(30)  M(u,  v)  du  4-  N(u,  v)  dv  =  0. 

The  curves  defined  by  the  former  equation  are  called  integral  curves 
of  equation  (30).  From  the  integral  equation  we  get  equation  (29) 
by  differentiation.  It  must  be  possible  then  to  obtain  equation  (30) 


FAMILIES  OF  CURVES  79 

from  the  integral  equation  and  (29).  But  c  does  not  appear  in 
(29),  consequently  the  latter  equation  differs  from  (30)  by  a  factor 

at  most.     Hence  'M—  —  N  —  =  0.    Suppose,  now,  that  we  have 
dv  du  o  .  o  , 

another  integral  of  (30),  as  ^Hw,  v)  =  e.    Then  M  -^-  —  N -^-  =  0. 

dv  cu 

The  elimination  of  M  and  N  from  these  equations  gives     ^  ^  =  0 ; 

d(u,v) 

from  which  it  follows  that  ty  is  a  function  of  </>.  Moreover,  ^r  can 
by  any  function  of  <£.  But  we  have  seen  that  if  ^  is  a  function 
of  <£,  the  families  of  curves  </>  =  const,  and  -fy  =  const,  are  the  same. 
Hence  all  integrals  of  equation  (30)  of  the  form  <f>=c  or  ^  =  e 
define  the  same  family  of  curves.  However,  equation  (30)  may 
admit  of  an  integral  in  which  the  constant  of  integration  enters 
implicitly,  as  F(u,  t>,  c)  =  0.  But  if  this  be  solved  for  <?,  we  obtain 
one  or  more  integrals  of  the  form  (27).  Hence  an  equation  of  the 
form  (30)  defines  one  family  of  curves  on  a  surface.  Although 
the  determination  of  the  curves  when  thus  defined  requires  the 
integration  of  the  equation,  the  direction  of  any  curve  at  a  point 
is  given  directly  by  means  of  (24). 

If  at  each  point  of  intersection  of  a  curve  Cl  with  the  curves 
of  a  family  the  tangents  to  the  two  curves  are  perpendicular  to 
one  another,  Cl  is  called  an  orthogonal  trajectory  of  the  curves.  Sup 
pose  that  the  family  of  curves  is  defined  by  equation  (30).  The 

7  r\ 

relation  between  the  ratios  —  and  —  >  which  determine  the  direc- 

du          ou 

tions  of  the  tangents  to  the  two  curves  at  the  point  of  intersection, 

r\  ~~\/T 

is  given  by  equation  (26).    If  we  replace  —  by ,  we  obtain 

cu  A 

(31)  (EN-  FM)  du  +  (FN—  GM)  dv  =  0. 

But  any  integral  curve  of  this  equation  is  an  orthogonal  trajectory 
of  the  given  curves.  Hence  a  family  of  curves  admits  of  a  family 
of  orthogonal  trajectories.  They  are  defined  by  equation  (31), 
when  the  differential  equation  of  the  curves  is  in  the  form  (30). 
But  when  the  family  is  defined  by  a  finite  equation,  such  as  (27), 
the  equation  of  the  orthogonal  trajectories  is 

(32) 


80  LINEAR  ELEMENT  OF  A  STJKFACE 

As  an  example,  we  consider  the  family  of  circles  in  the  plane  with  centers 
on  the  x-axis  whose  equation  is 

(i)  x2  +  y2  -  2  ux  -  a2, 

where  u  is  the  parameter  of  the  family  and  a  is  a  constant.  In  order  to  find  the 
orthogonal  trajectories  of  these  curves,  we  take  the  lines  x  =  const.  ,  y  =  const. 
for  parametric  curves,  in  which  case 

E  =  G  =  1,        F  =  0, 

3? 
and  write  the  equation  (i)  in  the  form  (27),  thus 

x  +  i  (y2  -  a2)  =  2  u. 

x 

Now  equation  (32)  is  2  xy  dx  —  (x2  —  y2  +  a2)  dy  =  0,  of  which  the  integral  is 


where  v  is  the  constant  of  integration.  Hence  the  orthogonal  trajectories  are  circles 
whose  centers  are  on  the  y-axis. 

An  ordinary  differential  equation  of  the  second  degree,  such  as 
(33)  H  (u,  v)  du2  +  2  S(u,  v)  du  dv  +  T(u,  v)  dv2  =  0, 

is  equivalent  to  two  equations  of  the  first  degree,  which  are  found 
by  solving  this  equation  as  a  quadratic  in  dv.  Hence  equation  (33) 
defines  two  families  of  curves  upon  the  surface.  We  seek  the  con 
dition  that  the  curves  of  one  family  be  the  orthogonal  trajectories 
of  the  other,  or,  in  other  words,  the  condition  that  (33)  be  the  equa 
tion  of  an  orthogonal  system,  as  previously  defined.  If  &x  and  Jc2 

denote  the  two  values  of  —  -  obtained  from  (33),  we  have 

du 


From  (26)  it  follows  that  the  condition  that  the  two  directions  at 
a  point  corresponding  to  KI  and  KZ  be  perpendicular  is 

E  +  FK  +  tc    +  GK      =  0. 


If  the  above  values  are  substituted  in  this  equation,  we  have 
the  condition  sought;  it  is 

(34) 


MINIMAL  CURVES  81 

35.  Minimal  curves  on  a  surface.  An  equation  of  the  form  (33) 
is  obtained  by  equating  to  zero  the  first  fundamental  form  of 
a  surface.  This  gives 

Edv?+  ZFdudv  +  Gdv2=  0, 

and  it  defines  the  double  family  of  imaginary  curves  of  length 
zero  which  lie  on  the  surface.  In  this  case  equation  (34)  reduces 
to  JSG  —  F2=0;  hence  the  minimal  lines  on  a  surface  form  an 
orthogonal  system  only  when  the  surface  is  an  isotropic  develop 
able  (§  31). 

An  important  example  of  these  lines  is  furnished  by  the  system 
on  the  sphere.  If  we  take  a  sphere  of  unit  radius  and  center  at 
the  origin,  its  equation,  x2+  y*+  z2  =  l,  can  be  written  in  either  of 
the  forms 


1  —  z       x  -f  iy 


where  u  and  v  denote  the  respective  ratios,  and  evidently  are  conju 
gate  imaginaries.  If  these  four  equations  are  solved  for  z,  ?/,  z,  we  find 

u  -h  v  i(v  —  u)  uv  —  1 

s**      -,  ' A  i   i  -.--,       \ f_  A»  ,•—  9 

~uv+l  uv +\  uv+1 

From  these  expressions  we  find  that  the  linear  element,  in  terms 
of  the  parameters  u  and  v^  is  given  by 

,o       4:dudv 


(36) 


(1  + 


Hence  the  curves  u  =  const,  and  v  —  const,  are  the  lines  of  length 
zero. 

Eliminating  u  from  the  first  two  and  the  last  two  of  equations 
(35),  we  get 

x  4-  (1  —  v*)y  —  2  iv  =  0, 


(37) 

1  i(i?  +  l)z  +  2vy+i(l-vz)  =  0. 

Hence  all  the  points  of  a  curve  v  =  const,  lie  on  the  line 

—  vz)Y—  2iv  =  Q, 


82  LINEAR  ELEMENT  OF  A  SURFACE 

where  X,  F,  Z  denote  current  coordinates.    In  consequence  of  (35), 
these  equations  can  be  written 

X-x,       Y-y.       Z-z, 


where  #0,  y^  z0  are  the  coordinates  of  a  particular  point.    In  like 
manner  the  curves  u  =  const,  are  the  minimal  lines 

X-x.        Y-y.    ,=  Z-z, 


EXAMPLES 

1 .  Show  that  the  most  general  orthogonal  system  of  circles  in  the  plane  is  that 
of  the  example  in  §  34. 

2.  Show  that  on  the  right  conoid  x  =  ucosv,  y  =  usinv,  z  =  au,  the  curves 
dw2  —  (w2  -f  a2)  dv2  =  0  form  an  orthogonal  system. 

3.  When  the  coefficients  of  the  linear  elements  of  two  surfaces, 

ds2  =  Erfu*  +  2  Fidudv  +  Gidv2,     ds*  =  E2du?  +  2  F2dudv  +  G2di;2, 

are  not  proportional,  and  points  with  the  same  curvilinear  coordinates  on  each  of 

the  surfaces  are  said  to  correspond,  there  is  a  unique  orthogonal  system  on  one 

surface  corresponding  to  an  orthogonal  system  on  the  other;  its  equation  is 

(Fi^a  -  FzEi)du*  +  (EaGi  -  EiGz)  dudv  +(GiFz  -  G2Fi)du2  =  0. 

4.  If  61  and  02  are  solutions  of  the  equation 

a^ 

—  —  At/  _  u, 


dag/3      2     da     8? 
where  X  is  any  function  of  a  and  /J,  the  equations 


+*5  3  2*  . 

define  a  surface  referred  to  its  minimal  lines. 

36.  Variation  of  a  function.  Let  S  be  a  surface  referred  to  any 
system  of  coordinates  ?/,  v,  and  let  <j>  (w,  v)  be  a  function  of  u  and  v. 
When  the  values  of  the  coordinates  of  a  point'Tlif  of  the  surface  are 
substituted  in  <£,  we  obtain  a  number  c  ;  and  consequently  the  curve 

(38) 


VABIATION  OF  A  FUNCTION  83 

passes  through  M.  In  a  displacement  from  M  along  this  curve  the 
value  of  (f>  remains  the  same,  but  in  any  other  direction  it  changes 
and  the  rate  of  change  is  given  by 

d±      d$k 

dc)  du       dv 


where  k  —  dv/du  determines  the  direction.  As  thus  written  it  is 
understood  that  the  denominator  of  the  right-hand  member  is 
positive. 

For  the  present  we  consider  the  absolute  value  of  ~-t  and  write 

ds 


(39) 


ds 


du       dv 


where  e  is  ±  1  according  as  the  sign  of  the  numerator  is  positive  or 
negative.  The  minimum  value  of  A  is  zero  and  corresponds  to  the 
direction  along  the  curve  (38).  In  order  to  find  the  maximum  value 
we  equate  to  zero  the  derivative  of  A  with  respect  to  Jc.  This  gives 


From  (32)  it  follows  that  this  value  of  k  determines  the  direction 
at  right  angles  to  the  tangent  to  $  =  c  at  the  point.  By  substituting 
this  value  of  k  in  (39)  we  get  the  maximum  value  of  A.  Hence: 

The  differential  quotient  -^-  of  a  function  <f>  (w,  v)  at  a  point  on  a 

ds 

surface  varies  in  value  with  the  direction  from  the  point.  It  equals 
zero  in  the  direction  tangent  to  the  curve  <f>=  c,  and  attains  its  greatest 
absolute  value  in  the  direction  normal  to  this  curve,  this  value  being 


m^\-ZF-z-z-+G 


dv  du  dv 


<4°>  ;        S 

A  means  of  representing  graphically  the  magnitude  of  the  differ 
ential  quotient  A  for  any  direction  is  given  by  the  following  theorem : 

If  in  the  tangent  plane  to  a  surface  at  a  point  M  the  positive  half 
tangents  at  M,  corresponding  to  all  values  of  k,  positive  and  negative, 


84  LINEAR  ELEMENT  OF  A  SURFACE 

be  drawn,  and  on  them  the  corresponding  lengths  A  be  laid  off  from  M, 
the  locus  of  the  extremities  of  these  lengths  is  a  circle  tangent  to  the 
curve  <£=  const. 

The  proof  of  this  theorem  is  simplified  if  we  effect  a  transfor 
mation  of  curvilinear  coordinates.  Thus  we  take  for  the  new  coor 
dinate  lines  the  curves  (f>  —  const,  and  their  orthogonal  trajectories. 
We  let  the  former  be  denoted  by  uv  —  const,  and  the  latter  by 
vl  =  const.,  and  indicate  by  subscript  1  functions  in  terms  of  these 
parameters.  Now  Fl=  0,  so  that 


-t  J.    —  7 

where  \  denotes  the  value  of  dvjdu^  which  determines  a  given 
direction,  and  the  maximum  length  is  (J&\)~*.    From  (23)  we  have 


cos      =     .  »          sin      = 


where  6Q  is  the  angle  which  the  given  direction  makes  with  the 
tangent  to  the  curve  vl=  const.  Hence  if  we  regard  the  tangents 
at  M  to  the  curves  vl  —  const,  and  ul  =  const,  as  axes  of  coordinates 
in  the  tangent  plane,  the  coordinates  of  the  end  of  a  segment  of 
length  A  are 


The  distance  from  this  point  to  the  mid-point  of  the  maximum 
segment,   measured  along  the   tangent  to  vt=  const.,   is  readily 

found  to  be  -  =<>  which  proves  the  theorem. 


37.  Differential  parameters  of  the  first  order.    If  we  put 

» 


(41)  A^  = 

equation  (40)  can  be  written 


(3)'-** 


where  now  the  differential  quotient  corresponds  to  the  direction 
normal  to  the  curve  <>  =  const.     The  left-hand  member  of  this 


DIFFERENTIAL  PARAMETERS  85 

equation  is  evidently  independent  of  the  nature  of  the  parameters 
u  and  v  to  which  the  surface  is  referred.  Consequently  the  same 
is  true  of  the  right-hand  member.  Hence  A^  is  unchanged  in 
value  when  there  is  any  change  of  parameters  whatever.  The 
full  significance  of  this  result  is  as  follows.  Given  a  new  set  of 
parameters  defined  by  M=/I(MI,  i^),  v=/2(w1,  v^\  let  ^(u^  vj 
denote  the  result  of  substituting  these  expressions  for  u  and  v  in 
<£  (w,  v),  and  write  the  linear  element  thus  : 

ds2  =  El  du*  +  2Fl  duldvl  +  Gl  dv*. 

The  invariance  of  A^  under  this  transformation  is  expressed  by 
the  identical  equation  , 


EG-F 


We  leave  it  to  the  reader  to  verify  this  directly  with  the  aid  of 
equations  (9).    The  invariant  A^  is  called  the  differential  parame 
ter  of  the  first  order  ;  this  name  and  the  notation  are  due  to  Lame.* 
Consider  for  the  moment  the  partial  differential  equation 

(42)  A^  =  0 

and  a  solution  </>  =  const.    From  the  latter  we  get,  by  differentiation, 

d<l>  ,     ,  3$  ,       A 
—  -  du  -f  —  dv  =  0. 

du  dv 

O  J  O    J 

If  we  replace  —  and  —  in  (42)  by  dv  and  —  du,  which  are  evi 
dently  proportional  to  them,  we  obtain 

Edu*+  2  Fdudv  +  Gdv*=  0. 


Hence  the  integral  curves  of  equation  (42)  are  lines  of  length  zero, 
and  conversely  if  (/>  =  const,  is  a  line  of  length  zero,  the  function  </> 
is  a  solution  of  equation  (42). 

Another  particular  case  is  that  in  which  A^  is  a  function  of  <£,  say 

(43)  A,*  =  *•<*). 

*  Lemons  sur  les  coordonnees  curvilignes  et  leurs  diverses  applications,  p.  5.  Paris,  1859. 


86  LINEAR  ELEMENT  OF  A  SURFACE 

From  (41)  it  is  seen  that  when  we  put 


equation  (43)  becomes 

(44)  A1l9=l. 

As  denned,  6  is  a  function  of  $;  hence  the  family  of  curves 
6  —  const,  is  the  same  as  the  family  </>  =  const.  Suppose  we  have 
such  a  family,  and  we  take  the  curves  9  —  const,  for  the  curves 
u  =  const,  and  their  orthogonal  trajectories  for  v  =  const.,  thus 
effecting  a  change  of  parameters.  Since  Aj%=l,  it  follows  from 
(41)  that  ^  =  1,  and  consequently  the  linear  element  is 

(45)  ds2=du?  +  Gdv'2. 

Since  now  the  linear  element  of  a  curve  v  —  const,  is  du,  the  length 
of  the  curve  between  its  points  of  intersection  with  two  curves 
u  =  u0  and  u  =  u^  is  u^  —  UQ.  Moreover,  this  length  is  the  same  for 
the  segment  of  every  curve  v  —  const,  between  these  two  curves. 
For  this  reason  the  latter  curves  are  said  to  be  parallel.  Con 
versely,  in  order  that  the  curves  u  =  const,  of  an  orthogonal  sys 
tem  be  parallel,  it  is  necessary  that  the  linear  element  of  the 
curves  v  =  const,  be  independent  of  v.  Hence  E  must  be  a  func 
tion  of  u  alone,  which,  by  a  transformation  of  coordinates,  can  be 
made  equal  to  unity.  Hence  we  have  the  theorem  : 

A  necessary  and  sufficient  condition  that  the  curves  of  a  family 
(/>  =  const,  be  parallel  is  that  \(f>  be  a  function  of  <f>. 

Let  (f)  =  const,  and  ^  =  const,  be  the  equations  of  two  curves 
upon  a  surface,  through  a  point  M,  and  let  6  denote  the  angle 
between  the  tangents  at  M.  If  we  put 


E  _  F  +  G 

dv  dv  \dv  du       du  dv  /          du  du 

(46)  \(*,*)=  --  EG  -I" 

the  expression  (21)  for  cos  6  can  be  written 

(47)  cos, 


DIFFERENTIAL  PARAMETERS  87 

Since  cos  6  is  an  invariant  for  transformations  of  coordinates,  it 
follows  from  this  equation  that  Ax(0,  ^r)  also  is  an  invariant.  It  is 
called  the  mixed  differential  parameter  of  the  first  order.  An  imme 
diate  consequence  of  (47)  is  that 

A,(*.  * )  =  ° 

is  the  condition  of  orthogonality  of  the   curves  (f>  =  const,  and 
i/r  =  const. 

Now  equation  (22)  can  be  written 


r  \du  dv       dv  c 
which  by  means  of  the  function  <e)  (w,  v),  defined  thus  by  Darboux,* 

can  be  written  in  the  abbreviated  form 
(49)  sin  6  = 


Since  all  the  functions  in  this  identity  except  ®  (<^,  ^r)  are  known 
to  be  invariants,  we  have  a  proof  that  it  also  is  an  invariant.  It 
is  a  mixed  differential  parameter  of  the  first  order.  From  (47) 
and  (49)  it  follows  that 

(50)  A,2  (<£,  f  )  +  ©2  ((/>,  t)  -  A^  •  A^  ; 

consequently  the  three  invariants  denned  thus  far  are  not  inde 
pendent  of  one  another. 

From  (41)  and  (46)  it  follows  that 

ri  _   Tf<  jfi 

^U  =  W        A^'^=l^'        AlV  =  ^' 

and  from  these  we  find 

(51)  02  (u,  v)  =  A,H  •  V  -  Ai2(«.  »)  =  ^  ' 
Consequently 


@2(w,  t;)  ©2(w,  v) 

Hence  i£,  ^,  and  G1  are  differential  invariants  of  the  first  order. 
*  Lemons,  Vol.  Ill,  p.  197. 


88  LINEAR  ELEMENT  OF  A  SURFACE 

Another  result  of  these  equations  is  the  following.    If  the  param 
eters  of  the  surface  are  changed  in  accordance  with  the  equations 

Ui  =  Ui(u,  v),         v^v^u,  v), 
and  the  resulting  linear  element  is  written, 

ds*  =  El  du*  +  2  Fl  du^dv^  +  6^  dv*, 
the  value  of  El  is  given  by 


and  7^  and  6^  are  found  in  like  manner.  In  consequence  of  (51) 
these  equations  are  equivalent  to  (13),  which  were  found  by  direct 
calculation. 

38.  Differential  parameters  of  the  second  order.  Thus  far  we 
have  considered  differential  invariants  of  the  first  order  only.  We 
introduce  now  one  of  the  second  order,  discovered  by  Beltrami.* 
To  this  end  we  study  the  integral 

n  = 

for  an  ordinary  portion  of  the  surface  bounded  by  a  closed  curve  C 
(cf.  §  33).  For  convenience  we  put 

Gz±_Fd±  Ez±_F3± 

*  du          dv  dv          du 

(53)  M=  -  -  -  ,         N=  -  —  -  , 

so  that,  in  consequence  of  (46),  we  have 


This  may  be  written 


If  we  apply  Green's  theorem  to  the  first  integral,  this  equation 

reduces  to 

(54)          n=  C(j>(Mdv-Ndu)-  ff$(^+(j^}dudv> 

*Ricerche  di  analisi  applicata  alia  geometria,  Giornale  di  matematiche,  Vol.  II 
(1864),  p.  365. 


DIFFERENTIAL  PARAMETERS  89 

where  the  first  integral  is  curvilinear  and  is  taken  about  C  in  the 
customary  manner.  Evidently  du  and  dv  refer  to  a  displacement 
along  C.  If  we  indicate  by  8  variations  in  directions  normal  to  C 
and  directed  toward  the  interior  of  the  contour,  then  from  (23) 
and  (25)  it  follows  that 


Edu  +  Fdv  _-H8v  F  du  +  G  dv  __ 

~dT  ~ST  ds  8s 


Hence  M  dv  — 

du  Ss       dv  8*/  8s 


All  of  the  terms  in  this  equation,  with  the  exception  of  —  (  -  —  h  -r—  )  > 

H\du       dv  J 

are  independent  of  the  choice  of  parameters.  Hence  the  latter  is 
an  invariant.  It  is  called  the  differential  parameter  of  the  second 
order  and  is  denoted  by  A2i/r.  In  consequence  of  (53)  we  have 


..... 

(56) 


In  the  foregoing  discussion  it  has  been  assumed  that  only  real 
quantities  appear.  But  all  these  results  can  be  obtained  directly 
from  algebraic  considerations  of  quadratic  differential  forms  * 
without  any  hypothesis  regarding  the  character  of  the  variables  ; 
hence  the  differential  parameters  can  be  used  for  any  kind  of 
curvilinear  coordinates. 

In  addition  to  A2c/>  there  are  other  differential  invariants  of  the 
second  order,  such  as 


And          AA  Q,  i/r),         A,  (A^,  A^),         ©  (A 

are  mixed  invariants  of  the  second  order.    In  like  manner  we  can 
find  a  group  of  invariants  of  the  third  order  ;  for  instance, 

AAM>,    AA(4>.M>)>    A.A,*,    A  A*.  •••• 

*  Cf.  Bianchi,  Lezioni  di  geometria  differ  enziale,  Vol.  I,  chap.  ii.   Pisa,  1902. 


90  LINEAR  ELEMENT  OF  A  SURFACE 

These  invariants  and  others,  which  can  be  obtained  by  an  evident 
extension  of  this  method,  involve  functions  c/>,  A/T,  .  •  •  ,  E,  F,  G,  and 
their  derivatives. 

Conversely,  we  shall  show  *  that  every  invariant  of  the  form 

T     ftw   v  r   dE  dG  *W  I    ^ 

/==/(A  ^   tr,   —  »   •  •  -,  —  »   •  •  -,  0,  -^-,   •  •  -,  i/r,  -^L,   •  •  •), 

dw  di>  dw  du 

where  <£,  -^,  .  .  .  are  independent  functions,  is  expressible  by  means 
of  the  symbols  A  and  ©.  Already  we  have  seen  that  E,  F,  and  G 
can  be  expressed  in  terms  of  Axw,  Axv,  and  A1(w,  v).  Moreover,  from 
(48)  it  follows  that 


when  X  is  any  function  whatever.    Hence  all  the  terms  in  /  can  be 
expressed  in  terms  of  the  symbols  A  and  ®,  applied  to 


Since  u  and  v  do  not  appear  explicitly  in  /,  we  can  effect  a  change 
of  parameters,  replacing  u  and  v  by  </>  and  ty  respectively,  and  con 
sequently  we  express  /  in  terms  of  (/>,  ^,  •  •  •,  and  the  differential 
invariants  obtained  by  applying  the  operators  A  and  <*)  to  these 
functions.  In  case  $  is  the  only  function  appearing  in  /,  we  can 
take  for  i/r,  in  the  change  of  parameters,  any  invariant  of  c/>,  such  as 
A^  or  A2<£,  so  long  as  it  is  not  a  function  of  (/>,  E,  F>  or  G. 

EXAMPLES 

4 

1  .  When  the  linear  element  of  a  surface  is  in  the  form 

ds2  =  \(du^  +  dv^), 

where  X  is  a  function  of  u  and  D,  both  u  and  v  are  solutions  of  the  equation  A20  =  0, 
the  differential  parameter  being  formed  with  respect  to  the  right-hand  member. 

2.  Show  that  on  the  surface 

x  =  u  cos  u,         y  =  u  sin  v,        z  =  av  -f-  0  (u), 
the  curves  it  =  const,  are  parallel. 

3.  When  the  linear  element  is  in  the  form 

ds2  =  cos^adu2  +  sin2  a:  eh?2, 
where  a  is  a  function  of  u  and  u,  both  u  and  v  are  solutions  of  the  equation 


*  Cf.  Beltrami,  I.e.,  p.  357. 


SYMMETRIC  COORDINATES  91 

4.  If  the  curves  0  =  const.,  \p  =  const,  form  an  orthogonal  system  on  a  surface, 
the  projection  on  the  x-axis  of  any  displacement  on  the  surface  is  given  by 

dx    d\b        dx    dd> 
dx  =  - £=  + 2= , 


*    A0 

where  ds  and  da-  are  the  elements  of  length  of  the  curves  0  =  const.,  ^  =  const. 
respectively. 

5.  If  /and  0  are  any  functions  of  u  and  u,  then 


.         a/  a0         i  if  a0     £/3«AA  .     ,     a/a0. 

,  0)  =  ^-  --^  AIU  +  (^     -  +  -     -  A!  (w,  u)  +  ^  ^  Aii>, 
du  du  \cu  cv       cv  du/  cv  cv 

A2/  =  ^A2w  +  ^A2u  +  ^AlW  +  2^-  Ai(u,  v)  +  ^AiU. 

CM  CU  SU2  0ttCtJ  SV2 

39.  Symmetric  co'drdinates.  We  have  seen  that  through  every 
point  of  a  surface  there  pass  two  minimal  curves  which  lie  entirely 
on  the  surface,  and  that  these  curves  are  defined  by  the  differential 
equation  Edv?+2  Fdudv  +  G  dv2  =  0. 

If  the  finite  equations  of  these  curves  be  written 

a  (w,  v)  =  const.,          fi  (w,  v)  =  const., 
it  follows  from  (42)  that 
(5T)  A,  («)=<),         A1(/3)  =  0. 

Since  for  any  parameters 

/^«\        w 
= 

when  the  curves  a  ~  const.,  ft  =  const.,  are  taken  as  parametric, 
the  corresponding  coefficients  E  and  G  are  zero,  and  consequently 
the  linear  element  of  the  surface  has  the  form 

(59)  ds2  =  \  dad/3, 

where,  in  general,  X  is  a  function  of  a  and  {3.  Conversely,  as  fol 
lows  from  (58),  when  the  linear  element  has  the  form  (59)  equa 
tions  (57)  are  satisfied  and  the  parametric  curves  are  minimal. 
Hence  the  only  transformations  of  coordinates  which  preserve  this 
form  of  the  linear  element  are  those  which  leave  the  minimal  lines 
parametric,  that  is 

(60)  or     a  =  - 


92  LINEAR  ELEMENT  OF  A  SURFACE 

where  F  and  Fl  are  arbitrary  functions.  Whenever  the  linear  ele 
ment  has  the  form  (59),  we  say  that  the  parameters  are  symmetric. 
The  above  results  are  given  by  the  theorem : 

When  a  and  ft  are  symmetric  coordinates  of  a  surface,  any  two 
arbitrary  functions  of  a  and  ft  respectively  are  symmetric  coordi 
nates,  and  they  are  the  only  ones. 

The  general  linear  element  of  a  surface  can  be  written  as  the 
product  of  two  factors,  namely 

(61)  d**: 

If  t  and  t1  denote  integrating  factors  of  the  respective  terms  of  the 
right-hand  member  of  this  equation,  a  pair  of  symmetric  coordinates 
is  given  by  the  quadratures 


(62) 


When  these  values  are  substituted  in  (61),  and  the  result  is  com 

pared  with  (59),  it  is  seen  that  X  =  — 

ttl 

The  first  of  equations  (62)  can  be  replaced  by 

,      da  ^FiH      da>' 

=—  >  t 

du 


Eliminating  t  from  these  equations,  we  have 

E^-F^ 

dv         du       .  ccc, 

<63>  -  IT     =lTu 

If   this    equation    be    multiplied  by   :   ~Z     »   the   result   can   be 
reduced  to 

r*-o£ 

dv          cu       .  dec 


ISOTHERMIC  PARAMETERS  93 

From  these  equations  it  follows  that 


or,  by  (56), 

(65)  A2tf  =  0. 

It  is  readily  found  that  /3  also  satisfies  this  condition. 

40.  Isothermic  and  isometric  parameters.  When  the  surface  is 
real,  and  the  coordinates  also,  the  factors  in  (61)  are  conjugate 
imaginary.  Hence  the  conjugate  imaginary  of  t  can  be  taken 
for  tr  In  this  case  a  and  fi  are  conjugate  imaginary  also.  In 
what  follows  we  assume  that  this  choice  has  been  made,  and  write 

(66)  a  =  <£+ty,         £  =  </>  —  iyfr. 
If  these  values  be  substituted  in  (59),  we  get 

(67)  ds*  =  \(d<t>2+d^). 

At  once  we  see  that  the  curves  c/>  =  const,  and  ^r  =  const,  form 
an  orthogonal  system.  Moreover,  the  elements  of  arc  of  these 
lines  are  V\d-*fr  and  ^\d(f>  respectively.  Consequently  when  the 
increments  d<f>  and  d^  are  taken  equal,  the  four  points  (<£,  i/r), 
(<f)  -f  c?<£,  i/r),  ($,  i/r  -f  efo/r),  (<£  -f  tity,  ^  -f  eityr)  are  the  vertices  of  a 
small  square.  Hence  the  curves  (f>  =  const,  and  ^|r  =  const,  divide 
the  surface  into  a  network  of  small  squares.  On  this  account 
these  curves  are  called  isometric  curves,  and  <f>  and  ty  isometric 
parameters.  These  lines  are  of  importance  in  the  theory  of  heat, 
and  are  termed  isothermal  or  isothermic,  which  names  are  used 
in  this  connection  as  synonymous  with  isometric. 

Whenever  the  linear  element  can  be  put  in  the  symmetric  form, 
equations  similar  to  (66)  give  at  once  a  set  of  isometric  parameters. 
And  conversely,  the  knowledge  of  a  set  of  isometric  parameters  leads 
at  once  to  a  set  of  symmetric  parameters.  But  we  have  seen  that  when 
one  system  of  symmetric  parameters  is  known,  all  the  others  are 
given  by  equations  of  the  form  (60).  Hence  we  have  the  theorem : 

Given  any  pair  of  real  isometric  parameters  <£,  -v/r  for  a  surface  ; 
every  other  pair  <£x,  ty1  is  given  by  equations  of  the  form 


where  F  and  FQ  are  any  functions  conjugate  imaginary  to  one  another. 


94  LINEAR  ELEMENT  OF  A  SURFACE 

Consider,  for  instance,  the  case 

(68)  *1+^1  =  ^(0+i». 
From  the  Cauchy-Riemann  differential  equations 

(69)  ?*i  =  *±i,          ?&=_?*i, 
d<l>       c^  3^          8<f> 

it  follows  that  (f)l  and  ^l  are  functions  of  both  <£  and  T/T.  Hence 
the  curves  </>1  =  const.,  ^1  =  const,  are  different  from  the  system 
<f>  —  const.,  T/T  =  const.  Similar  results  hold  when  +  i  is  replaced 
by  —  i  in  the  argument  of  the  right-hand  member  of  (68).  Hence 

There  is  a  double  infinity  of  isometric  systems  of  lines  upon  a  sur 
face;  when  one  system  is  known  all  the  others  can  be  found  directly. 

If  the  value  (66)  for  a  be  substituted  in  the  first  of  equations  (57), 
the  resulting  equation  is  reducible  to 


Since  <f>  and  ^r  are  real,  this  equation  is  equivalent  to 

(70)  A^A.VT,      A1(^,f)  =  o. 

From  (58)  it  is  seen  that  these  equations  are  the  condition  that 
E  —  G,  F=  0,  when  <f>  and  i/r  are  the  parameters.    Hence  equations 

(70)  are  the  necessary  and  sufficient  conditions  that  $  and  i/r  be 
isometric  parameters. 

Again,  when  a  in  (65)  is  replaced  by  </>+  i^r,  and  all  the  func 
tions  are  real,  we  have  f 

(71)  A2*=0, 


Conversely,  when  we  have  a  function  (f>  satisfying  the  first  of  these 
equations,  the  expression 


cu          cv  ,  on          ov  , 

dv 


is  an  exact  differential.    Call  it  d^r  ;  then 

jr^  —  E—  G  —  —  F— 

du  _  dv^_c^r  du  c)v 

H  ~~du'  H 


ISOTHERMIC  ORTHOGONAL  SYSTEMS  95 

If  these  equations  be  solved  for  — »  — »  we  get 

du     dv 

/r_ox  dv  du      dd>  dv  du       d<f> 

(<o)  =  —  ">  =  —  • 

H  du  H  dv 

When  we  express  the  condition  —  ( — )  =  —  ( — )  >  we  find  that 

dv\duj      du\dvj 

A2-»/r=0.  Moreover,  these  two  functions  (f>  and  ^  satisfy  (70), 
in  consequence  of  (72)  and  (73),  and  therefore  they  are  isometric 
parameters.  Hence : 

A  necessary  and  sufficient  condition  that  <f>  be  the  isometric  param 
eter  of  one  family  of  an  isometric  system  on  a  surface  is  that  A2c/>  =  0 ; 
the  isometric  parameter  of  the  other  family  is  given  by  a  quadrature. 

Incidentally  we  remark  that  if  u  and  v  are  a  pair  of  isometric 
parameters,  equations  (72)  and  (73)  reduce  to  (69). 

41.  Isothermic  orthogonal  systems.  If  the  linear  element  of  a 
surface  is  given  in  the  form  (67)  and  the  parameters  are  changed 
in  accordance  with  the  equations 


the  linear  element  becomes 


where  the  accents  indicate  differentiation.  However,  this  trans 
formation  of  parameters  has  not  changed  the  coordinate  lines  ; 
the  coefficients  are  now  no  longer  equal,  but  in  the  relation 

<»>  i-f 

where  U  and  V  denote  functions  of  u  and  v  respectively. 

Conversely,  when  this  relation  is  satisfied  the  linear  element 
may  be  written 


and  by  the  transformation  of  coordinates, 
(75)  4>  =  C^/lfdu,         ^  =  C 


96  LINEAR  ELEMENT  OF  A  SURFACE 

it  is  brought  to  the  form  (67),  whatever  be  U  and  V\  and  the  coor 
dinate  lines  are  unaltered.    Hence  : 

A  necessary  and  sufficient  condition  that  an  orthogonal  system  of 
parametric  lines  on  a  surface  form  an  isothermic  system  is  that  the 
coefficients  of  the  corresponding  linear  element  satisfy  a  relation  of 
the  form  (74). 

We  seek  now  the  necessary  and  sufficient  condition  which  a 
function  o>  (w,  v)  must  satisfy  in  order  that  the  curves  o>  =  const. 
and  their  orthogonal  trajectories  form  an  isothermic  system. 
Either  o>,  or  a  function  of  it,  is  the  isothermic  parameter  of  the 
curves  o>  =  const.  We  denote  this  parameter  by  </>;  then  </>=/(«)  . 
Since  </>  must  be  a  solution  of  equations  (71),  we  have,  on  substitution, 

(76)  A2o>  ./'(G>)  +  \a>  ./"(a>)  =  0, 

where  the  primes  indicate  differentiation  with  respect  to  &>.    If  this 
equation  is  written  in  the  form 


we  see  that  the  ratio  of  the  two  differential  parameters  is  a  func 
tion  of  co.  Conversely,  if  this  ratio  is  a  function  of  o>,  the  function 
/(a>),  obtained  by  two  quadratures  from 

(77)  /'(*>)  =  *-/£>, 

will  satisfy  equations  (71).    Hence: 

A  necessary  and  sufficient  condition  that  a  family  of  curves 
a)  =  const,  and  their  orthogonal  trajectories  form  ''  an  isothermic  sys 
tem  is  that  the  ratio  of  A2&>  and  AjO)  be  a  function  of  &>. 

Suppose  we  have  such  a  function  w  ;  then  the  orthogonal  tra 
jectories  of  the  curves  &>  =  const,  can  be  found  by  quadrature  ;  for, 
the  differential  equation  of  these  trajectories  is 


(78^  \~  dv     -  du/ "-  '  \~  dv     -  Su 

If  equation  (76)  be  written  in  the  form 

0  I  *f>          dv          du         v   i  „, .          wu          t/i/         n 
—    f  (&))  — — — — —   H —  I  r  (ft))  —————   =  u, 

O  I    *^          \  /  TT-  *  r\  I    4/  \  /  -TT- 

<7V 


ISOTHEEMIC  ORTHOGONAL  SYSTEMS  97 

it  is  seen  that  an  integrating  factor  of  equation  (78)  is  f'(a))/H, 
where  f'(co)  is  given  by  (77).  Hence  /(«)  and  the  function  <f, 
obtained  by  the  quadrature 

__#&>      ^cto  s^fo      -rJo<* 


are  a  pair  of  isometric  parameters.    From  these  equations  and  (77) 
it  follows  that 


and  consequently,  by  means  of  (52),  the  linear  element  can  be 
given  the  form 

(80)  ds*  =  —  (da*  +  «8/sSd"  ctyA 

A^  x 

The  linear  element  of  the  plane  referred  to  rectangular  axes  is  ds2  =  dx2  4-  dy2. 
Consequently  x  and  y  are  isothermic  parameters,  and  we  have  the  theorem  : 

The  plane  curves  whose  equations  are  obtained  by  equating  to  constants  the  real 
and  imaginary  parts  of  any  function  of  x  +  iy  or  x  -  iy  form  an  isothermal  orthog 
onal  system  ;  and  every  such  system  can  be  obtained  in  this  way. 

c2 

For  example,  consider  0  4-  ty  =  --  —  » 

x  —  iy 

where  c  is  any  constant.    From  this  it  follows  that 


x2  4-  yz  x2  4-  y2 

Hence  the  circles  0  =  const.,  $  =  const,  form  an  isothermal  orthogonal  system, 
and  0  and  ^  are  isothermic  parameters. 

The  above  system  of  circles  is  a  particular  case  of  the  system  considered  in  §  34. 
We  inquire  whether  the  latter  also  form  an  isothermal  system.    If  we  put 

u  =  x  4-  i  (2/2  -  «2), 

1  2d) 

we  find  that  AIO>  =  —  (w2  4-  4  a2),         A^u  =  — —  • 

x2  x2 

Hence  the  ratio  of  AIW  and  A2W  is  a  function  of  w,  and  consequently  the  system  of 
circles  is  isothermal.    From  (77)  it  follows  that  the  isothermic  parameter  of  the 

first  family  is  0  =  — tan-1  —  ,  and  the  parameter  of  the  orthogonal  family  is 
2  a  2  a 

1  w  x2  4-  a2 

\b  —  —  tanh-1  —  >        w  =  y  4 • 

2  a  2  a  y 


98  LINEAR  ELEMENT  OF  A  SURFACE 


EXAMPLES 

1.  Show  that  the  meridians  and  parallels  on  a  sphere  form  an  isothermal  orthog 
onal  system,  and  determine  the  isothermic  parameters. 

2.  Show  that  a  system  of  confocal  ellipses  and  hyperbolas  form  an  isothermal 
orthogonal  system  in  the  plane. 

3.  Show  that  the  surface 


x  _      I  (a2  -  u)  (a^v)       y  _      I  (b*  -  u)  (b2  -  v)        z  _      ! 
a  ~  \  (a2  -  &2)  (a2  -  c2) '     b  ~  \  (62  -  a2)  (62  -  c2) '     c  ~~  \ 


(C2  _  U)  (C2  _ 


f(c2-a2)(c2-62) 
is  an  ellipsoid,  and  that  the  parametric  curves  form  an  isothermal  orthogonal  system. 

4.  Find  the  curves  which  bisect  the  angles  between  the  parametric  curves  on 
the  surface  %  _  u  +  v        y  _  u  _  v  _  uv 

a~     2  b~"~2~  =2' 

and  show  that  they  form  an  isothermal  orthogonal  system. 

5.  Determine  0  (v)  so  that  on  the  right  conoid  x  —  u  cos  v,  y  =  u  sin  v,  z  =  <f>  (v) 
the  parametric  curves  form  an  isothermal  orthogonal  system,  and  show  that  the 
curves  which  bisect  the  angles  between  the  parametric  curves  form  a  system  of 
the  same  kind. 

6.  Express  the  results  of  Ex.  4,  page  82,  in  terms  of  the  parameters  0  and  ^ 
defined  by  (66). 

42.  Conformal  representation.  When  a  one-to-one  correspond 
ence  of  any  kind  is  established  between  the  points  of  two  sur 
faces,  either  surface  may  be  said  to  be  represented  on  the  other. 
Thus,  if  we  roll  out  a  cylindrical  surface  upon  a  plane  and  say 
that  the  points  of  the  surface  correspond  to  the  respective  points 
of  the  plane  into  which  they  are  developed,  we  have  a  representa 
tion  of  the  surface  upon  the  plane.  Furthermore,  as  there  is  no 
stretching  or  folding  of  the  surface  in  this  development  of  it  upon 
the  plane,  lengths  of  lines  and  the  magnitude  of  angles  are  unal 
tered.  It  is  evidently  impossible  to  make  such  a  representation  of 
every  surface  upon  a  plane,  and,  in  general,  two  surfaces  of  this 
kind  do  not  admit  of  such  a  representation  upon  one  another. 
However,  it  is  possible,  as  we  shall  see,  to  represent  one  surface 
upon  another  in  such  a  way  that  the  angles  between  correspond 
ing  lines  on  the  surfaces  are  equal.  In  this  case  we  say  that  one 
surface  has  conformal  representation  on  the  other. 

In  order  to  obtain  the  condition  to  be  satisfied  for  a  conformal 
representation  of  two  surfaces  S  and  Sr,  we  imagine  that  they  are 
referred  to  a  corresponding  system  of  real  lines  in  terms  of  the 


CONFOKMAL  KEPKESENTATION  99 

same  parameters  w,  v,  and  that  corresponding  points  have  the  same 
curvilinear  coordinates.  We  write  their  linear  elements  in  the 
respective  forms 

ds2  =  Edu*+  2  Fdudv  +  G  dv2,    ds'2  =  £'du*+  2  F'dudv  +  G'dv*. 

Since  the  angles  co  and  to'  between  the  coordinate  lines  at  corre 
sponding  points  must  be  equal,  it  is  necessary  that 

F  F' 


(81) 


y/EG 


If  00  and  0'Q  denote  the  angles  which  a  curve  on  S  and  the  corre 
sponding  curve  on  Sr  respectively  make  with  the  curves  v  =  const. 
at  points  of  the  former  curves,  we  have,  from  (23)  and  (25), 

.    n        H    dv  .     .        Q         H  du 

sin  00  =  -—  —  ,  sin  (to  -  00)  =  —  =  — 

ds 


.    a,       H'    dv  •    ,   ,     ai\       H'   du 

sm  6'0  =  —=—,         sin  (to1  -  6[)  =  -=  — 
s'  V  £'<&?' 


By  hypothesis  a>r=±a)  and  6[  —  ±  0Q,  according  as  the  angles  have 
the  same  or  opposite  sense.    Hence  we  have 

H'  —  -      —  —  H'  —  -       H  du 

~  ds  '  ~ 


according  to  the  sense  of  the  angles.    From  these  equations  we  find 


which,  in  combination  with  (81),  may  be  written 


where  t2  denotes  the  factor  of  proportionality,  a  function  of  u  and 
v  in  general.    From  (83)  it  follows  at  once  that 

(84)  ds'*=t 


* 


And  so  when  the  proportion  (83)  is  satisfied,  the  equations  (81) 
and  (82)  follow.    Hence  we  have  the  theorem : 

A  necessary  and  sufficient  condition  that  the  representation  of  two 
surfaces  referred  to  a  corresponding  system  of  lines  be  conformal  is 


100        LINEAR  ELEMENT  OF  A  SUKPACE 

that  the  first  fundamental  coefficients  of  the  two  surfaces  be  propor 
tional,  the  factor  of  proportionality  being  a  function  of  the  param 
eters  ;  the  representation  is  direct  or  inverse  according  as  the  relative 
positions  of  the  positive  half  tangents  to  the  parametric  curves  on  the 
two  surfaces  are  the  same  or  symmetric. 

Later  we  shall  find  means  of  obtaining  conformal  representations. 

From  (84)  it  follows  that  small  arcs  measured  from  correspond 
ing  points  on  S  and  Sf  along  corresponding  curves  are  in  the  same 
ratio,  the  factor  of  proportionality  being  in  general  a  function  of 
the  position  of  the  point.  Conversely,  when  the  ratio  is  the  same 
for  all  curves  at  a  point,  there  is  a  relation  such  as  (84),  with  t  a 
function  of  u  and  v  at  most.  And  since  it  holds  for  all  directions, 
we  must  have  the  proportion  (83).  On  this  account  we  may  say 
that  two  surfaces  are  represented  conformally  upon  one  another 
when  in  the  neighborhood  of  each  pair  of  homologous  points  corre 
sponding  small  lengths  are  proportional. 

43.  Isometric  representation.  Applicable  surfaces.  When  in  par 
ticular  the  factor  t  is  equal  to  unity,  corresponding  small  lengths 
are  equal  as  well  as  angles.  In  this  case  the  representation  is  said 
to  be  isometric,  and  the  two  surfaces  are  said  to  be  applicable.  The 
significance  of  the  latter  term  is  that  the  portion  of  one  surface  in 
the  neighborhood  of  every  point  can  be  so  bent  as  to  be  made  to 
coincide  with  the  corresponding  portion  of  the  other  surface  with 
out  stretching  or  duplication.  It  is  evident  that  such  an  applica 
tion  of  one  surface  upon  another  necessitates  a  continuous  array  of 
surfaces  applicable  to  both  S  and  $r  This  process  of  transformation 
is  called  deformation,  and  Sl  is  called  a  deform  of  S  and  vice  versa. 
An  example  of  this  is  afforded  by  the  rolling  of  a  cylinder  on 
a  plane. 

Although  a  conformal  representation  can  be  established  between 
any  two  surfaces,  it  is  not  true,  as  we  shall  see  later,  that  any  two 
surfaces  admit  of  an  isometric  representation  upon  one  another. 
From  time  to  time  we  shall  meet  with  examples  of  applicable  sur 
faces,  and  in  a  later  chapter  we  shall  discuss  at  length  problems 
which  arise  concerning  the  applicability  of  surfaces.  However, 
we  consider  here  an  example  afforded  by  the  tangent  surface  of  a 
twisted  curve. 


APPLICABLE  SURFACES  101 

We  recall  that  if  #,  y,  z  are  the  coordinates  of  a  point  on  the 
curve,  expressed  in  terms  of  the  arc,  the  equations  of  the  surface  are 
of  the  form       f  =  x  +  ^     v  =  y  +  y't,     £=z  +  z% 
and  the  linear  element  of  the  surface  is 

d <r*  =  /I  +  -\  dsz  +  2  dsdt  +  dt\ 

where  p  denotes  the  radius  of  curvature  of  the  curve. 

Since  this  expression  does  not  involve  the  radius  of  torsion,  it 
follows  that  the  tangent  surfaces  to  all  curves  which  have  the 
same  intrinsic  equation  p  =f(s)  are  applicable  in  such  a  way  that 
points  on  the  curves  determined  by  the  same  value  of  s  correspond. 
As  there  is  a  plane  curve  with  this  equation,  the  surface  is  appli 
cable  to  the  plane  in  such  a  way  that  points  of  the  surface  corre 
spond  to  points  of  the  plane  on  the  convex  side  of  the  plane  curve. 

The  tangents  to  a  curve  are  the  characteristics  of  the  osculating 
planes  as  the  point  of  osculation  moves  along  the  curve,  and  con 
sequently  they  are  the  axes  of  rotation  of  the  osculating  plane  as 
it  moves  enveloping  the  surface.  Instead  of  rolling  the  plane  over 
the  tangent  surface,  we  may  roll  the  surface  over  the  plane  and  bring 
all  of  its  points  into  coincidence  with  the  plane.  It  is  in  this  sense 
that  the  surface  is  developable  upon  a  plane,  and  for  this  reason 
it  is  called  a  developable  surface  (cf.  §  27).  Later  it  will  be  shown 
that  every  surface  applicable  to  the  plane  is  the  tangent  surface  of 
a  curve  (§  64). 

44.  Conformal  representation  of  a  surface  upon  itself.  We  return 
to  the  consideration  of  conformal  representation,  and  remark  that 
another  consequence  of  equations  (83)  is  that  the  minimal  curves 
correspond  upon  S  and  Sr.  Conversely,  when  two  surfaces  are 
referred  to  a  corresponding  system  of  lines,  if  the  minimal  lines  on 
the  two  surfaces  correspond,  equations  (83)  must  hold.  Hence : 

A  necessary  and  sufficient  condition  that  the  representation  of  two 
surfaces  upon  one  another  be  conformal  is  that  the  minimal  lines 
correspond. 

If  the  minimal  lines  upon  the  two  surfaces  are  known  and  taken 
as  parametric,  the  linear  elements  are  of  the  form 

(85)  ds2  =  X  dadfr         ds'2  =  \  da^dftv 


102  LINEAR  ELEMENT  OF  A  SURFACE 

Hence  a  conformal  representation  is  defined  in  the  most  general 
way  by  the  equations 

W  «l  =  F(a),          ft  =  *;(£), 

or 

(87)  ^  =  F(ft),         ft  =  *;(«), 

where  F  and  F1  are  arbitrary  functions  which  must  be  conjugate 
imaginary  when  the  surfaces  are  real. 

Instead  of  interpreting  (85)  as  the  linear  elements  of  two  sur 
faces  referred  to  their  minimal  lines,  we  can  look  upon  them  as 
the  linear  element  of  the  same  surface  in  terms  of  two  sets  of 
parameters  referring  to  the  minimal  lines.  From  this  point  of 
view  equations  (86)  and  (87)  define  the  most  general  conformal 
representation  of  a  surface  upon  itself.  If  we  limit  our  considera 
tion  to  real  surfaces  and  put,  as  before, 

a  =  $  +  i^,     £  =  <£-ty,     a1=<^1+i>1,     ft=01-*^1, 
the  functions  fa  i/r  and  fa,  ^  are  pairs  of  isothermic  parameters. 
Now  equations  (86),  (87)  may  be  written 

(88)  <#>1+i>1  =  7^±^). 
Consequently  we  have  the  theorem : 

When  a  pair  of  isothermic  parameters  fa  ty  of  a  surface  are  known 
and  the  surface  is  referred  to  the  lines  <j>  =  const.,  ^r  =  const.,  the 
most  general  conformal  representation  of  the  surface  upon  itself  is 
obtained  by  making  a  point  (fa  \fr)  correspond  to  the  point  (fa,  i^), 
into  which  it  can  be  transformed  in  accordance  with  equation  (88). 

As  a  corollary  of  this  theorem,  we  have : 

When  a  pair  of  isothermic  parameters  is  known  for  each  of  two 
surfaces,  all  the  conformal  representations  of  one  surface  upon  the 
other  can  be  found  directly. 

Consider  two  pairs  of  isothermic  parameters  fa  ty  and  fa,  ^  for 
a  surface  S,  and  suppose  their  relation  is 

(89)  &+*+!  =  F(t  +  i+). 

If  two  curves  C  and  Cl  are  in  correspondence  in  this  representa 
tion,  their  parametric  equations  must  be  the  same  functional  rela 
tion  between  the  parameters,  namely, 

*,)  =0. 


CONFORMAL  REPRESENTATION  103 

Denote  by  9  and  0l  the  angles  which  C  and  C^  make  with  the 
curves  ^  =  const,  and  ^1  =  const,  respectively.  If  we  write  the 
linear  element  of  S  in  the  two  forms 


it  follows  from  (23)  that 


a  deb  .    a 

cos  0  =  y      =  ,         sin  0  = 


cos  0  =  -  --  =,       sin      = 


From  these  expressions  we  derive  the  following 


=  ^ 

d(f>  —  i  c?i/r 

so  that  in  consequence  of  (89)  we  have 
(90)  «..«,-„=  :*^ 


where  7^0  is  the  function  conjugate  to  7^,  and  the  accents  indicate 
differentiation  with  respect  to  the  argument.  If  T  and  Fx  are 
another  pair  of  corresponding  curves,  and  their  angles  are  denoted 
by  6  and  0V  it  follows  from  (90)  that 


,, 
OI> 


For,  the  right-hand  member  of  (90)  is  merely  a  function  of  the 
position  of  the  point  and  is  independent  of  directions.  Hence  in 
any  conformal  representation  defined  by  an  equation  of  the  form 
(89)  the  angles  between  corresponding  curves  have  the  same  sense. 
When,  now,  the  correspondence  satisfies  the  equation 


the  equation  analogous  to  (90)  is 


Hence  0l-0l=0-0'i 

consequently  the  corresponding  angles  are  equal  in  the  inverse  sense. 


104  LINEAB  ELEMENT  OF  A  SURFACE 

45.  Conformal  representation  of  the  plane.  For  the  plane  the 
preceding  theorem  may  be  stated  thus  : 

The  most  general  real  conformal  representation  of  the  plane  upon 
itself  is  obtained  by  making  a  point  (x,  y)  correspond  to  the  point 
(x^  y^),  where  x^iy^  is  any  function  of  x  +  iy  or  x  —  iy. 


We  recall  the  example  of  §  41,  namely 

0)  Xl  +  iyi  =  ;rrfc' 

where  c  is  a  real  constant.    This  equation  is  equivalent  to 


and  also  to 

C2X 


Hence  the  parallels  x  =  const,  and  y  —  const.,  in  the  xy-plane,  are  represented 
in  the  z^-plane  by  circles  which  pass  through  the  origin  and  have  their  centers 
on  the  respective  axes.  Conversely,  these  circles  in  the  xy-plane  correspond  to 
the  parallels  in  the  Xi^/i-plane. 

If  we  put  „        „ 

o;2  +  y*  =  r2,         x*  +  y*  =  rf, 

equations  (ii)  and  (iii)  may  be  written 

<*>  ?-?•   f-S-   — • 

Hence  corresponding  points  are  on  the  same  line  through  the  origin,  and  their 
distances  from  it  are  such  that  rr\  =  c2.  On  this  account  equations  (iv)  are 
said  to  define  an  inversion  with  respect  to  the  circle  x2  +  y2  =  c2,  or,  since  TI  =  c2/r, 
o  transformation  by  reciprocal  radii  vector -es. 

From  §  44  it  follows  that  corresponding  angles  are  equal  in  the  inverse  sense. 

For  the  case 

c2 

(v)  xi  +  iy\  = — 

x  +  iy 

the  equations  analogous  to  (iv)  are 

-  =  ?±,        V  =  -Vl. 
r      n'        r          ri* 

Hence  the  point  PI  (xi,  y\)  corresponding  to  P  (x,  y)  lies  on  the  line  which  is  the 
reflection  in  the  x-axis  of  the  line  OP,  and  at  the  distance  OPi  =  c2/r.  Evidently 
this  transformation  is  the  combination  of  an  inversion  and  the  transformation 

*i  =  *,  y\  =  -  y- 

One  finds  that  the  transformations  (i)  and  (v)  have  the  following  properties : 

Every  straight  line  is  transformed  into  a  circle  which  passes  through  the  origin ; 
and  conversely. 

Every  circle  which  does  not  pass  through  the  origin  is  transformed  into  a  circle. 


CONFOKMAL  REPRESENTATION  105 

We  propose  now  the  problem  of  finding  the  most  general  con- 
formal  transformation  of  the  plane  into  itself,  which  changes 
circles  not  passing  through  the  origin  into  circles.  In  solving  it 
we  refer  the  plane  to  symmetric  parameters  #,  fi,  where 

a  =  x  -f-  iy,          f$—x  —  iy. 

The  equation  of  any  circle  which  does  not  pass  through  the 
origin  is  of  the  form 

(91)  ca(S+  aa  +  5/3  +  d  =  0, 

where  <z,  5,  c,  d  are  constants  ;  when  the  circle  •  is  real  a  and  b 
must  be  conjugate  imaginaries  and  c  real.  Equation  (91)  defines  @ 
as  a  function  of  a.  If  we  differentiate  the  equation  three  times 
with  respect  to  #,  and  eliminate  the  constants  from  the  resulting 
equations,  we  find 

(92)  3/3"2-2/3'/3'"=0, 

where  the  accent  indicates  differentiation  with  respect  to  a. 
Moreover,  as  equation  (91)  contains  three  independent  constants, 
it  is  the  general  integral  of  (92). 

We  know  that  the  most  general  conformal  representation  of 
the  plane  upon  itself  is  given  by 

(93)  a1  =  A(a),          ft  =  £(£), 
or 

(94)  «!  =  £(£),         13,  =  A  (a). 

Our  problem  reduces,  therefore,  to  the  determination  of  functions 
A  and  B,  such  that  the  equation 

(95)  3  ft'2-  2  ft'  ft"  =0, 

where  the  accent  indicates  differentiation  with  respect  to  av  can 
be  transformed  by  (93)  or  (94)  into  (92). 

We  consider  first  equations  (93),  which  we  write 


Now 

ff^*!L*pta_ 

*     30    da    da 

In  like  manner  we  find  ft'  and  ft".    When  their  values  are  sub 
stituted  in  (95)  we  get,  since  A(  and  B'  are  different  from  zero, 


3  ft"2  -  2  ffff"  +  -     (3  B"2  -  2  B'B'")  £'4  +        (3  A'»  -  2  A[Al")  ft'2  =  0. 
B  Al 


106  LINEAR  ELEMENT  OF  A  SURFACE 

Since  equation  (95)  must  be  directly  transformable  into  (92),  it 

follows  that 

(96)  3  £"2-  2  />"£'"  =  0,         3  A^~  2A[A™  =  0. 

As  these  equations  are  of  the  form  (92),  their  general  integrals 
are  similar  to  (91).  Hence  the  most  general  forms  of  (93)  for 
our  problem  are 

•> 


Moreover,  when,  these  values  are  substituted  in  an  equation  in 
a^  (Sl  of  the  form  (91),  the  resulting  equation  in  a  and  ft  is  of 
this  form. 

Equation  (91)  may  likewise  be  looked  upon  as  defining  a  in 
terms  of  ft,  so  that  a,  as  a  function  of  ft,  satisfies  an  equation  of  the 
form  (92)  ;  similarly  for  al  as  a  function  of  ftr  Hence  if  we  had  used 
(94),  we  should  have  been  brought  to  results  analogous  to  (97)  ;  and 
therefore  the  most  general  forms  of  (94)  for  our  problem  are 

(98)  «i=!4±T'          ft-54**- 

bs/3+b^  «,«  +  «, 

Hence  : 

When  'a  plane  is  defined  in  symmetric  parameters  a,  ft,  the  most 
general  conformal  representation  of  the  plane  upon  itself,  for  which 
circles  correspond  to  circles  or  straight  lines,  is  given  by  (97)  or  (98).* 

EXAMPLES 

1.  Deduce  the  equations  which  define  the  most  general  conformal  representation 
of  a  surface  with  the  linear  element  cZs2  =  dv?  +  (a2  —  u^dv2  upon  itself. 

2.  Show  that  the  surfaces 

x  —  u  cos  v,        y  =  u  sin  u,        z  =  au, 

x  —  u  cos  v,         y  —  u  sin  v,         z  =  a  cosh  -*  -  , 

are  applicable.  Find  the  curve  in  which  a  plane  through  the  z-axis  cuts  the  latter 
surface,  and  deduce  the  equations  of  the  conformal  representation  of  these  surfaces 
on  the  plane. 

3.  When  the  representation  is  defined  by  (97),  what  are  the  coordinates  of  the 
center  and  radius  of  the  circle  in  the  <n-plane  which  corresponds  to  the  circle  of 
center  (c,  d)  and  radius  r  in  the  or-plane  ? 

*  The  transformations  (97)  and  (98)  play  an  important  role  in  the  theory  of  functions. 
For  a  more  detailed  study  of  them  the  reader  is  referred  to  the  treatises  of  Picard,  Darboux, 
and  Forsyth. 


SURFACES  OF  REVOLUTION  107 

4.  Show  that  in  the  conformal  representation  (97)  there  are,  in  general,  two 
distinct  points,  each  of  which  corresponds  to  itself ;  also  that  if  7  and  5  are  the 
values  of  a  at  these  points,  then 


K  = 


ai  —  d      a  —  5  ai  +  ai  +  V(ai  —  a4)2  +  4  a2a3 

5  .  Find  the  condition  that  the  origin  be  the  only  point  which  corresponds  to  itself, 
and  show  that  if  the  quantities  01,  ag,  ^3,  a±  are  real,  a  circle  in  the  a-plane  through 
the  origin  0  corresponds  to  a  circle  in  the  a^plane  through  0  and  touching  the  other 
circle  ;  also  that  a  circle  touching  the  x-axis  at  0  corresponds  to  itself. 

6.  The  equation  2  ai  =  (a  —  b)  a  -f  -  -  ?  where  a  and  6  are  constants,  defines  a 

conformal  representation  of  the  plane  upon  itself,  such  that  circles  about  the  origin 
and  straight  lines  through  the  latter  in  the  a-plane  correspond  to  confocal  ellipses 
and  hyperbolas  in  the  ai-plane. 

7.  In  the  conformal  representation  «i  =  logo:  to  lines  parallel  to  the  x-  and 
y-axes  in  the  ai-plane  there  correspond  lines  through  the  origin  and  circles  about 
it  in  the  a-plane,  and  to  any  orthogonal  system  of  straight  lines  in  the  ai-plane 
an  orthogonal  system  of  logarithmic  spirals  in  the  a-plane. 

46.  Surfaces  of  revolution.  By  definition  a  surface  of  revolution 
is  the  surface  generated  by  a  plane  curve  when  the  plane  of  the 
curve  is  made  to  rotate  about  a  line  in  the  plane.  The  various 
positions  of  the  curve  are  called  the  meridians  of  the  surface,  and 
the  circles  described  by  each  point  of  the  curve  in  the  revolution 
are  called  the  parallels.  We  take  the  axis  of  rotation  for  the  2-axis, 
and  for  o>axis  and  ?/-axis  any  two  lines^perpendicular  to  one  another, 
and  to  the  z-axis,  and  meeting  it  in  the  same  point.  For  any  posi 
tion  of  the  plane  the  equation  of  the  curve  may  be  written  z  =  </>(r), 
Avhere  r  denotes  the  distance  of  a  point  of  the  curve  from  the  2-axis. 
We  let  v  denote  the  angle  which  the  plane,  in  any  of  its  positions, 
makes  with  the  #2-plane.  Hence  the  equations  of  the  surface  are 

(99)  x  =  r  cos,v,          y  =  rsinv,         z=(f>(r). 


The  linear  element  is 

(100)  ds2  =  [1  +  <£'2  (r)]  dr'2  + 

If  we  put 

a01) 

the  linear  element  is  transformed  into 
(102) 


108  LINEAR  ELEMENT  OF  A  SURFACE 

where  X  is  a  function  of  u,  which  shows  that  the  meridians  and 
parallels  form  an  isothermal  system.  As  this  change  of  parameters 
does  not  change  the  parametric  lines,  the  equations 

x  =  u,         y  =  v, 

define  a  conformal  representation  of  the  surface  of  revolution  upon 
the  plane  in  which  the  meridians  and  parallels  correspond  to  the 
straight  lines  x  =  const,  and  y  =  const,  respectively. 

By  definition  a  loxodromic  curve  on  a  surface  of  revolution  is  a 
curve  which  cuts  the  meridians  under  constant  angle.  Evidently 
it  is  represented  on  the  plane  by  a  straight  line.  Hence  loxodromic 
curves  on  a  surface  of  revolution  (99)  are  given  by 


C-  Vl  +  $* 


+  bv  +  c  =  0, 


where  a,  £,  c  are  constants. 

Incidentally  we  have  the  theorem  : 

When  the  linear  element  of  a  surface  is  reducible  to  the  form 


where  \  is  a  function  of  u  or  v  alone,  the  surface  is  applicable  to  a 
surface  of  revolution. 

For,  suppose  that  X  is  a  function  of  u  alone.  Put  r  =  Vx  and 
solve  this  equation  for  u  as  a  function  of  r.  If  the  resulting 
expression  be  substituted  in  (101),  we  find,  bya  quadrature,  the 
function  <f>(r)y  for  which  equations  (99)  define  the  surface  of 
revolution  with  the  given  linear  element.  r, 

When,  in  particular,  the  surface  of  revolution  is  the  unit  sphere, 
with  center  at  the  origin,  we  have 

r  =  sin  w,         z=  Vl  —  r2  =  cos  w, 

where  u  is  the  angle  which  the  radius  vector  of  the  point  makes 
with  the  positive  z-axis.    Now 

=  log  tan  |  . 

Hence  the  equations  of  correspondence  are 

,  u 

x  =  log  tan-,         y  =  v. 


MERCATOR  REPRESENTATION  109 

This  representation  is  called  a  Mercator  chart  of  the  sphere  upon 
the  plane.  It  is  used  in  making  maps  of  the  earth  for  mariners. 
A  path  represented  by  a  straight  line  on  the  chart  cuts  the  meridians 
at  constant  angle. 

47.  Conformal  representations  of  the  sphere.  We  have  found 
(§  35)  that  when  the  unit  sphere,  with  center  at  the  origin,  is 
referred  to  minimal  lines,  its  equations  are 

a  +  /3  •(£—«)  a/3-l 

(103)      "- 


where  a  and  j3  are  conjugate  imaginary.  Hence  the  parametric 
equation  of  any  real  circle  on  the  sphere  is  of  the  form 

'      ca{3+aa  +  b/3+d=Q, 

where  a  and  b  are  conjugate  imaginary  and  c  and  d  are  real. 
From  this  it  follows  that  the  problem  of  finding  any  conformal 
representation  of  the  sphere  upon  the  plane  with  circles  of  the 
former  in  correspondence  with  circles  or  straight  lines  of  the 
latter,  is  the  same  problem  analytically  as  the  determination  of 
this  kind  of  representation  of  the  plane  upon  itself.  Hence,  from 
the  results  of  §  45,  it  follows  that 

All  conformal  representations  of  the  sphere  (103)  upon  a  plane, 
with  circles  of  the  former  corresponding  to  circles  or  straight  lines 
of  the  latter,  are  defined  by 

a.a  +  a,,  .        bfi+b.  * 

<104)        ***-;{?+£     ^'y'=^A' 

We  wish  to  consider  in  particular  the  case  in  which  the  sphere 
is  represented  on  the  ^-plane  in  such  a  way  that  the  great  cir 
cle  determined  by  this  plane  corresponds  with  itself  point  for 
point. 

From  (103)  we  have  that  the  equations  of  this  circle  are 


*  The  representation  with  the  lower  signs  is  the  combination  of  the  one  with  the  upper 
sign  and  the  transformation  &i  =  /3,  /Si=  «,  which  from  (103)  is  seen  to  transform  a  figure 
bn  the  sphere  into  the  figure  symmetrical  with  respect  to  the  zz-plane. 


110  LINEAR  ELEMENT  OF  A  SURFACE 

When  these  values  are  substituted  in  (104)  it  is  found  that  we 
must  have  ,       »  r       i       A 

ai=«4»  bl=t>V  az=<*3=0Z=03=i), 

so  that  the  particular  form  of  (104)*  is  equivalent  to 

*1=|(«+/9),        y,  =  £(£-«)• 

From  these  equations  and  (103)  we  find  that  the  equations 
of  the  straight  lines  joining  corresponding  points  on  the  sphere 
and  plane  are  reducible  to 

X  Y  1-Z 


For  all  values  of  a  and  ft  these  lines  pass  through  the  point  (0,  0,  1). 
Hence  a  point  of  the  plane  corresponding  to  a  given  point  P  upon 
the  sphere  is  the  point  of  intersection  with  the  plane  of  the  line 
joining  P  with  the  pole  (0,  0,  1).  This  form  of  representation  is 
called  the  stereographic  projection  of  the  sphere  upon  the  plane. 
It  is  evident  that  a  line  in  the  plane  corresponds  to  a  circle  on 
the  sphere  ;  this  circle  is  determined  by  the  plane  of  the  pole  and 
the  given  line. 

We  will  close  this  chapter  with  a  few  remarks  about  the  con- 
formal  representation  of  the  sphere  upon  itself.  From  the  fore 
going  results  we  know  that  every  such  representation  of  the 
sphere  (103)  is  given  by  equations  of  similar  form  in  a^  ftv  where 
the  latter  are  given  by  (86)  or  (87),  and  that  for  conformal  repre 
sentations  with  circles  in  correspondence  al  and  ^  have  the  values 
(97)  or  (98). 

We  consider  in  particular  the  case 


a.a 


The  expressions  of  the  linear  elements  of  the  sphere  are  found 
to  be  reducible  to 

4  dad/3  4  da^ft,         4  dad  ft 


2 

~ 


*  Here  we  have  used  the  upper  signs  in  (104). 


STEREOGRAPHIC  PROJECTION  111 

Hence,  equations  (105)  define  an  isometric  representation  of  the 
sphere  upon  itself.  Since  angles  are  preserved  in  the  same  sense 
by  (105),  this  representation  may  be  looked  upon  as  determining 
a  motion  of  configuration  upon  the  sphere  into  new  positions 
upon  it.  The  stationary  points  in  the  general  motion,  if  there 
are  any,  correspond  to  values  of  a  and  /3,  which  are  roots  of 
the  respective  equations 

If  tl  and  £2  are  the  roots  of  the  former,  those  of  the  latter  are  —  l/^ 
and  — 1/£2.    Hence  there  are  four  points  stationary  in  the  motion; 
their  curvilinear  coordinates  are 
1 

-L\  /.  -*-    X  /    j  •"     \  I    J. 

Lni     ~ 


" 


From  (103)  it  is  seen  that  the  first  two  are  at  infinity,  and  the 
last  two  determine  points  on  the  sphere,  so  that  the  motion  is  a 
rotation  about  these  points.  If  the  z-axis  is  taken  for  the  axis  of 
rotation,  we  have  from  (103)  that  the  roots  of  (106)  must  be  oo  and 
0  ;  hence  #2=  «3=  0,  so  that  (105)  becomes 


If  the  rotation  is  real,  these  equations  must  be  of  the  form 


=  e 


where  o>  is  the  angle  of  rotation. 

EXAMPLES 

1.  Find  the  equations  of  the  surface  of  revolution  with  the  linear  element 
ds2  =  dw2  +  (a2  -  w2)du2. 

2.  Find  the  loxodromic  curves  on  the  surface 

i     i  u 
X  =  MCOSU,      y  =  usmv,      z  =  a  cosh-1-, 

and  find  the  equations  of  the  surface  when  referred  to  an  orthogonal  system  of 
these  curves. 

3.  Find  the  general  equations  of  the  conformal  representation  of  the  oblate 
spheroid  upon  the  plane. 

4.  Show  that  for  the  surface  generated  by  the  revolution  of  the  evolute  of 
the  catenary  about  the  base  of   the  latter  the  linear  element  is   reducible   to 
ds2  =  du"2  +  u  dv2. 


112  LINEAR  ELEMENT  OF  A  SURFACE 

5.  A  great  circle  on  the  unit  sphere  cuts  the  meridian  v  =  0  in  latitude  <x  under 
angle  a.    Find  the  equation  of  its  stereographic  projection. 

6.  Determine    the    stereographic    projection  of    the    curve   x  =  asinwcosw, 
y  =  acos2w,  z  —  asinw  from  the  pole  (0,  a,  0). 

GENERAL  EXAMPLES 

1.  When  there  is  a  one-to-one  point  correspondence  between  two  surfaces,  the 
cross-ratio  of  four  tangents  to  one  surface  at  a  point  is  equal  to  the  cross-ratio  of 
the  corresponding  tangents  to  the  other. 

2.  Given  the  paraboloid 

x  =  2awcosu,     y=2&Msinv,     z  =  2  w2(a  cos2u  +  6sin2u), 

where  a  and  b  are  constants.  Determine  the  equation  of  the  curves  on  the  surface, 
such  that  the  tangent  planes  along  a  curve  make  a  constant  angle  with  the  xy-plane. 
Show  that  the  generators  of  the  developable  2,  enveloped  by  these  planes,  make  a 
constant  angle  with  the  z-axis,  and  express  the  coordinates  of  the  edge  of  regression 
in  terms  of  v. 

3.  Find  the  orthogonal  trajectories  of  the  generators  of  the  surface  S  in  Ex.  2. 
Show  that  they  are  plane  curves  and  that  their  projections  on  the  xy-plane  are 
involutes  of  the  projection  of  the  edge  of  regression. 

4.  Let  C  be  a  curve  on  a  cone  of  revolution  which  cuts  the  generators  under 
constant  angle,  and  Ci  the  locus  of  the  centers  of  curvature  of  C.    Show  that  C\ 
lies  upon  a  cone  whose  elements  it  cuts  under  constant  angle. 

5.  When  the  polar  developable  of  a  curve  is  developed  upon  a  plane,  the  curve 
degenerates  into  a  point. 

6.  When  the  rectifying  developable  of  a  curve  is  developed  upon  a  plane,  the 
curve  becomes  a  straight  line. 

7.  Determine  <f>(o)  so  that  the  right  conoid, 

x  =  ucosv,     y=usinv,     z  =  (f>(v), 
shall  be  applicable  to  a  surface  of  revolution. 

8.  Determine  the  equations  of  a  conformal  representation  of  the  plane  upon 
itself  for  which  the  parallels  to  the  axes  in  the  ai-plane  correspond  to  lines  through 
a  point  (a,  b)  and  circles  concentric  about  it  in  the  a-plane. 

9.  The  equation  a\  =  c  sin  a,  where  c  is  a  constant,  defines  a  conformal  repre 
sentation  of  the  plane  upon  itself  such  that  the  lines  parallel  to  the  axes  in  the 
a-plane  correspond  to  confocal  ellipses  and  hyperbolas  in  the  ai-plane. 

10.  In  the  conformal  representation  of  the  plane  upon  itself,  given  by  ai  =  a2, 
to  lines  parallel  to  the  axes  in  the  ori-plane  there  correspond  equilateral  hyperbolas 
in  the  a-plane,  and  to  the  pencil  of  rays  through  a  point  in  the  ori-plane  and  the  cir 
cles  concentric  about  it  there  corresponds  a  system  of  equilateral  hyperbolas  through 
the  corresponding  point  in  the  or-plane  and  a  family  of  confocal  Cassini  ovals. 

11.  When  the  sides  of  a  triangle  upon  a  surface  of  revolution  are  loxodromic 
curves,  the  sum  of  the  three  angles  is  equal  to  two  right  angles. 

12.  The  only  conformal  perspective  representation  of  a  sphere  upon  a  plane  is 
given  by  (104). 


GENERAL  EXAMPLES  113 

13.  Show  that  equations  (105)  and  the  equations  obtained  from  (105)  by  the 
interchange  of  cc.  and  /3  define  the  most  general  isometric  representation  of  the 
sphere  upon  itself. 

14.  Let  each  of  two  surfaces  S,  S\  be  defined  in  terms  of  parameters  w,  u,  and 
let  points  on  each  with  the  same  values  of  the  parameters  correspond.    If  H  —  H\, 
where  the  latter  is  the  function  for  Si  analogous  to  H  for  S,  corresponding  elements 
of  area  are  equal  and  the  representation  is  said  to  be  equivalent.*  If  H  ^  HI  and 
the  parameters  of  S  are  changed  in  accordance  with  the  equations  u'  —  <f>  (w,  v), 
v'=  $  (u,  a),  the  condition  that  the  equations  u'  =  M,  v'=  v  define  an  equivalent  rep 
resentation  of  S  and  Si  is  H 


du  dv       cv  du      HI  (0,  \[<) 

15.  Under  what  conditions  do  the  equations 

x'  —  aix  +  azy  +  a3,        y'  =  b&  +  b2y  +  63 
define  an  equivalent  representation  of  the  plane  upon  itself  ? 

16.  Show  that  the  equations 


determine  an  equivalent  representation  of  the  surface  of  revolution  (99)  upon  the 
plane. 

17.  Given  a  sphere  and  circumscribed  circular  cylinder.    If  the  points  at  which 
a  perpendicular  to  the  axis  of  the  latter  meets  the  two  surfaces  correspond,  the 
representation  is  equivalent. 

18.  Find  an  equivalent  representation  of  the  sphere  upon  the  plane  such  that 
the  parallel  circles  correspond  to  lines  parallel  to  the  y-axis  and  the  meridians  to 
ellipses  for  which  the  extremities  of  one  of  the  principal  axes  are  (a,  0),  (—  a,  0). 

*  German  writers  call  it  "  flachentreu." 


CHAPTER  IV 


GEOMETRY  OF  A  SURFACE  IN  THE  NEIGHBORHOOD  OF  A  POINT 

48.  Fundamental  coefficients  of  the  second  order.  In  this  chapter 
we  study  the  form  of  a  surface  in  the  neighborhood  of  a  point  M 
of  it,  and  the  character  of  the  curves  which  lie  upon  the  surface 
and  pass  through  the  point.  We  recall  that  the  tangents  at  M  to 
all  these  curves  lie  in  a  plane,  —  the  tangent  plane  to  the  surface  at 
the  point. 

The  equation  of  the  tangent  plane  at  M(x,  y,  2),  namely  (II,  11), 
may  be  written 

(1)  (f- 

where  we  have  put 


H 


_  _ 

du  du 

dy^  dz 

~dv  dv 


H 


dz  dx 

du  du 

dz  dx 

do  dv 


H 


dx  dy 

dti  du 

dx  dy 

dv  dv 


We  define  the  positive  direction  of  the  normal  (§  25)  to  be  that 
for  which  the  functions  X,  I7,  Z  are  the  direction-cosines.  From  this 
definition  it  follows  that  the  tangents  to  the  curves  v  —  const,  and 
u  =  const,  at  a  point  and  the  normal  at  the  point  have  the  same 
mutual  orientation  as  the  #-,  ?/-,  and  2-axes. 

From  (2)  follow  the  identities 


(3) 


F        «-  fl 

^  7T  —  u» 

dv 


which  express  the  fact  that  the  normal  is  perpendicular  to  the  tan 
gents  to  the  coordinate  curves.  In  consequence  of  these  identities 
the  expression  for  the  distance  p  from  a  point  M '  (u  +  du,  v  •+•  dv) 
to  the  tangent  plane  at  M  is  of  the  second  order  in  du  and  dv. 
It  may  be  written 

(4)  p  =  ^X  dx  =  1  (D  du2  +2D'  dudv  +  &"  dv2)  +  e, 

lit 


COEFFICIENTS  OF  THE  SECOND  OKDEK 


115 


where  e  denotes  the  aggregate  of  terms  of  the  third  and  higher 
orders  in  du  arid  dv,  and  the  functions  Z>,  D\  D"  are  defined  by 


(5) 


dudv 


If  equations  (3)  be  differentiated  with  respect  to  u  and  v  respec 
tively,  we  get 


(6) 


dudv 


1  =  0, 

dv 


dv  du 
dX  dx  _  ~ 

dv  dv 


And  so  equations  (5)  may  be  written 


(7) 


'  _  V  Y  —  —  —        — 
~^^   dudv~     Ztdu  d 

,,_y     ^__y^^ 
**    -9*         ^  dv  dv 


dv  du 


The  quadratic  differential  form 
(8)  <1>  =  D  du*  +  2  D'dudv  -f  D"dv2 

is  called  the  second  fundamental  form  of  the  surface,  and  the  func 
tions  D,  D',  D"  the  fundamental  coefficients  of  the  second  order.  We 
leave  it  to  the  reader  to  show  that  these  coefficients,  like  those  of  the 
first  order,  are  invariant  for  any  displacement  of  the  surface  in  space. 

Later  we  shall  have  occasion  to  use  two  sets  of  formulas  which 
will  now  be  derived. 

From  the  equations  of  definition, 

toy 

~  I  ' 

cv) 
we  get,  by  differentiation  and  simple  reduction,  the  following : 

a^^_ia^  y^^  =  ?^_l?^, 

dE         v  dx  &x  _  l  dG 

~2  du' 


(9) 


^  \du]  '  ^  du  dv 


(10) 


dx 


dv  dudv 


2  dv 


116        GEOMETRY  OF  A  SURFACE  ABOUT  A  POINT 

Again,  if  the  expressions  (9)  be  substituted  in  the  left-hand  mem 
bers  of  the  following  equations,  the  reduced  results  may  be  written 
by  means  of  (2)  in  the  form  indicated : 


dv      "  du          \     du         du,  ' 


dv         du          \     dv          dv, 

Similar  identities  can  be  found  by  permuting  the  letters  x,  y,  z ; 
X,  F,  Z. 

From  the  fundamental  relation 

we  obtain,  by  differentiation  with  respect  to  u  and  v  respectively, 
the  identities 


These  equations  and  (7)  constitute  a  system  of  three  equations 

linear  in  — »  — »  — *  and  a  system  linear  in  — - ,  — ,  — -  •    Solving 

du     du    du  dv     dv     dv 

ij.  -y  O   -y 

for  —  and  for  — »  we  find,  by  means  of  (11), 

du  dv 

dX      FD'-GDdx  FD-ED' dx  ^ 

~du~         H* du  H*        dv' 

^dX  _FD"-GD'dx  FZ>'-RD"dx 

dv  "          H*         du  H*         dv 

7}  V  /}  7  ' 

The  expressions  for  —,•••,  --  are  obtained  by  replacing  x  by  y 

..     ,         du  dv 

and  z  respectively. 

By  means  of  these  equations  we  shall  prove  that  a  real  surface  whose  first  and 
second  fundamental  coefficients  are  in  proportion,  thus 

D      V      V" 
(14)  —  =  —  =  —  =—\ 

where  X  denotes  the  factor  of  proportionality,  is  a  sphere  or  a  plane.    We  assume 
that  the  minimal  lines  are  parametric.    In  consequence  we  have 

E  =  G  =  D  =  D"  =  0, 
so  that  equations  (13)  become 

dX -\—          —  —  \  — 

du         du  dv          dv 


RADIUS  OF  NORMAL  CURVATURE  117 

The  function  X  must  satisfy  the  condition 


dv  \    du/       du  \    dv 

which  reduces  to  ----  •  —  =  0.    Moreover,  we  have  two  other  equations  of 
dv  du      du  dv 

condition,  obtained  from  the  above  by  replacing  x  by  y  and  z  respectively.  Since 
the  proportion  to.ay  .  to  =  to  .0y  .  to 

du'  du    du      dv'  dv  '  dv 

is  not  possible  for  a  real  surface,  we  must  have  —  =  —  =  0  :  that  is,  X  is  a  con- 

du      dv 

stant.  When  X  is  zero  the  functions  X,  F,  Z  given  by  (15)  are  constant,  and 
consequently  the  surface  is  a  plane.  When  X  is  any  other  constant,  we  get, 
by  integration  from  (15), 

X  —  \x  +  a,         Y  =  \y  4-  6,         Z  =  Xz  +  c, 

where  a,  6,  c  are  constants.  From  these  equations  we  obtain  (\x  -f  a)2  4-  (\y  4-  6)2 
4-  (Xz  4-  c)2  —  1.  Since  this  is  the  general  equation  of  a  sphere,  it  follows  that  the 
above  condition  is  necessary  as  well  as  sufficient. 

v/49.  Radius  of  normal  curvature.  Consider  on  a  surface  S  any 
curve  C  through  a  point  M.  The  direction  of  its  tangent,  MT, 
is  determined  by  a  value  of  dv/du.  Let  o>  denote  the  angle  which 
the  positive  direction  of  the  normal  to  the  surface  makes  with  the 
positive  direction  of  the  principal  normal  to  C  at  Jf,  angles  being 
measured  toward  the  positive  binormal.  If  we  use  the  notation  of 
the  first  chapter,  and  take  the  arc  of  C  for  its  parameter,  we  have 


In  terms  of  —  and  —  the  derivatives  in  the   parenthesis  have 
,,      .  as  as 

the  forms 


fo  =  Zfa/duV 
aV  ~~  du2  \ds) 


du  dv  ds  ds      dv2  \ds 
so  that  the  above  equation  is  equivalent  to 

cos  w      D  du2  +  2  D'dudv  +  D"dv2 


(16) 


Edu*+  ZFdudv  4-  Gdv* 


As  the  right-hand  member  of  this  equation  depends  only  upon 
the  curvilinear  coordinates  of  the  point  and  the  direction  of  MT, 
it  is  the  same  for  all  curves  with  this  tangent  at  M.  Since  p  is 
positive,  the  angle  o>  cannot  be  greater  than  a  right  angle  for  one 
curve  tangent  to  MT,  if  it  is  less  than  a  right  angle  for  any  other 


118        GEOMETRY  OF  A  SURFACE  ABOUT  A  POINT 

curve  tangent  to  MT;  and  vice  versa.  We  consider  in  particular 
the  curve  in  which  the  surface  is  cut  by  the  plane  determined  by 
MT  and  the  normal  to  the  surface  at  M.  We  call  it  the  normal 
section  tangent  to  MT,  and  let  pn  denote  its  radius.  Since  the 
right-hand  member  of  equation  (16)  is  the  same  for  C  and  the 
normal  section  tangent  to  it,  we  have 

(17) 

P  Pn 

where  e  is  +1  or  —1,  according  as  w  is  less  or  greater  than  a  right 
angle;  for  p  and  pn  are  positive.  Equation  (17)  gives  the  follow 
ing  theorem  of  Meusnier: 

The  center  of  curvature  of  any  curve  upon  a  surface  is  the  pro 
jection  upon  its  osculating  plane  of  the  center  of  curvature  of  the 
normal  section  tangent  to  the  curve  at  the  point. 

In  order  to  avoid  the  ambiguous  sign  in  (17),  we  introduce  a 
new  function  R  which  is  equal  to  pn  when  0  <  o>  <  TT/%,  and  to  —  pn 
when  7r/Z<a><7r,  and  call  it  the  radius  of  normal  curvature  of  the 
surface  for  the  given  direction  MT.  As  thus  defined,  E  is  given  by 


R       Edu2  +  2  Fdudv  +  Gdv* 

Now  we  may  state  Meusnier's  theorem  as  follows  : 

If  a  segment,  equal  to  twice  the  radius  of  normal  curvature  for  a 
given  direction  at  a  point  on  a  surface,  be  laid  off  from  the  point  on 
the  normal  to  the  surface,  and  a  sphere  be  described  with  the  segment 
for  diameter,  the  circle  in  ivliich  the  sphere  is  met  by  the  osculating 
plane  of  a  curve  with  the  given  direction  at  the  point  is  the  circle  of 
curvature  of  the  curve. 

50.  Principal  radii  of  normal  curvature.  If  we  put  t  =  —  >  equa 
tion  (18)  becomes 

I      D+2D't+D"t 


When  the  proportion  (14)  is  satisfied,  R  is  the  same  for  all  values 
of  t,  being  oc  for  the  plane,  and  the  constant  —  1/X  for  the  sphere. 
For  any  other  surface  R  varies  continuously  with  t.  And  so  we 


PRINCIPAL  RADII  OF  CURVATURE  119 

seek  the  values  of  t  for  which  11  is  a  maximum  or  minimum. 
To  this  end  we  differentiate  the  above  expression  with  respect 
to  t  and  pnt  the  result  equal  to  zero.  This  gives 

(20)  (J}'+D"t)(E+2Ft+Gt2)-(F+Gt)(D  +  2D't  +  D"t2)  =  Q, 
or 

(21)  (FD"-GDl)t2+(FD"-GD)t+(ED'-FD)  =  (). 

Without  any  loss  of  generality  we  can  assume  that  the  parametric 
curves  are  such  that  E  =£  0,  so  that  we  have  the  identity 

(22)  (ED"-  GDf-  4  (FD"  —  DrG)  (ED'-FD) 

7T"2  T  2  F  H2 

=  4  —  (FD'-FDf+\  ED"—GD—--(ED'  —  FD)\. 
E  \_  E  J 

When  the  surface  is  real,  and  the  parameters  also,  the  right-hand 
member  of  this  equation  is  positive.  Since  the  left-hand  member 
is  the  discriminant  of  equation  (21),  the  latter  has  two  real  and 
distinct  roots.*  When  the  test  (III,  34)  is  applied  to  equation  (21), 
it  is  found  that  the  two  directions  at  a  point  determined  by  the 
roots  of  (21)  are  perpendicular.  Hence: 

At  every  ordinary  point  of  a  surface  there  is  a  direction  for  which 
the  radius  of  normal  curvature  is  a  maximum  and  a  direction  for 
which  it  is  a  minimum,  and  they  are  at  right  angles  to  one  another. 

These  limiting  values  of  R  are  called  the  principal  radii  of 
normal  curvature  at  the  point.  They  are  equal  to  each  other  for 
the  plane  and  the  sphere,  and  these  are  the  only  real  surfaces 
with  this  property. 

From  (20)  and  (19)  we  have 

D'+D"t_D  +  D't  _  1 
F+Gt        E  +  Ft~~R' 

Hence  the  following  relations  hold  between  the  principal  radii  and 
the  corresponding  values  of  t : 

f  E  +  Ft-R(D  +  D't)  =  Q, 
\F+Gt-R(D'  +  D"t)  =  0. 

*  In  order  that  the  two  roots  he  equal,  the  discriminant  must  vanish.  This  is  impos 
sible  for  real  surfaces  other  than  spheres  and  planes,  as  seen  from  (22).  For  an  imaginary 
surface  of  this  kind  referred  to  its  lines  of  length  zero,  we  have  from  (21)  that  D  or  D" 
is  zero,  since  F  ^  0.  The  vanishing  of  the  discriminant  is  also  the  necessary  and  sufficient 
condition  that  the  numerator  and  denominator  in  (19)  have  a  common  factor. 


120        GEOMETRY  OF  A  SURFACE  ABOUT  A  POINT 

When  t  is  eliminated  from  these  equations,  we  get  the  equation 
(24)     (DD"  -  D1'2)  A)2  -  (ED"  +  GD-2  FD')  R  +  (EG  -  F2)  =  0, 

whose  roots  are  the  principal  radii.    If  these  roots  be  denoted  by  pl 
and  /32,  we  have 


(25) 


!_      ^^ 


DD"-D'2 


PiP*  H 

Although  equations  (14)  hold  at  all  points  of  a  sphere  and  a 
plane,  and  for  no  other  surface,  it  may  happen  that  for  certain  par 
ticular  points  of  a  surface  they  are  satisfied.  At  such  points  R, 
as  given  by  (19),  is  the  same  for  all  directions,  and  the  equa 
tion  (21)  vanishes  identically.  When  points  of  this  kind  exist  they 
are  called  umbilical  points  of  the  surface. 

EXAMPLES 

1.  When  the  equation  of  the  surface  is  z  =f(x,  ?/),  show  that 

x,Y,z  =  ^J^, 

D,  D',  D"  =  — 


dz  dz  82z  d2z 

where  p  =•  —  >      a  —  —  >      r  =  —  »      s  —  — 

dx  dy  dx*  dxdy 

2.  Show  that  the  normals  to  the  right  conoid 


along  a  generator  form  a  hyperbolic  paraboloid. 

3.  Show  that  the  principal  radii  of  normal  curvature  of  a  right  conoid  at  a 
point  differ  in  sign. 

4.  Find  the  expression  for  the  radius  of  normal  curvature  of  a  surface  of  revolu 
tion  at  a  point  in  the  direction  of  the  loxodromic  curve  through  it,  which  makes  the 
angle  a  with  the  meridians. 

5.  Show  that  the  meridians  and  parallels  on  a  surf  ace  of  revolution,  x  =  u  cosu, 
y  =  u  sin  i>,  z  =  0  (w),  are  the  directions  in  which  the  radius  of  normal  curvature  is 
maximum  and  minimum  ;  that  the  principal  radii  are  given  by 


Pl         (1  +  0/2)  P2M 

and  that  />2  is  the  segment  of  the  normal  between  the  point  of  the  surface  and  the 
intersection  of  the  normal  with  the  z-axis. 

6.  Show  that  AIX  =  1  -  X2  and  AI  (x,  y)  =  -  XY,  where  the  differential  param 
eters  are  formed  with  respect  to  the  linear  element  of  the  surface. 


LINES  OF  CURVATUBE 


121 


51.  Lines  of  curvature.   Equations  of  Rodrigues.    We  have  seen 
that  the  curves  defined  by  equation  (21),  written 

(26)      (ED'  -  FD)  du2  +  (ED"  -  GD)  dudv  +  (FD"  -  GDf)  dvz=0, 

form  an  orthogonal  system.  As  defined,  the  two  curves  of  the  sys 
tem  through  a  point  on  the  surface  determine  the  directions  at  the 
point  for  which  the  radii  of  normal  curvature  have  their  maximum 
and  minimum  values.  These  curves  are  called 
the  lines  of  curvature,  and  their  tangents  at 
a  point  the  principal  directions  for  the  point. 
They  possess  another  geometric  property  which 
we  shall  now  find. 

The  normals  to  a  surface  along  a  curve 
form  a  ruled  surface.  In  order  that  the  sur 
face  be  developable,  the  normals  must  be 
tangent  to  a  curve  (§  27),  as  in  fig.  12.  If 
the  coordinates  of  a  point  Ml  on  the  normal 
at  a  point  M  be  denoted  by  xr  y^  z^  we  have 


FIG.  12 


where  r  denotes  the  length  MMr  If  M^  be  a  point  of  the  edge  of 
regression,  we  must  have 

dx+rdX+Xdr  _dy-{-r  dY+  Ydr  _dz  +  r  dZ  +  Zdr 
X  Y  Z 

Multiplying  the  numerators  and  the  denominators  of  the  respec 
tive  members  by  X,  F,  Z,  and  combining,  we  find  that  the  common 
ratio  is  dr.  Hence  the  above  equations  reduce  to 


or,  when  the  parametric  coordinates  are  used, 


(  8x  ,        dx  ,  dX  ,     ,  dX  , 

—  du  H d  v  +  r  I  —  du  H d  v 

du  dv  \du  dv 


(27) 


dv 
fa 

'dv 


du 


dv 


du  dv 


122        GEOMETRY  OF  A  SURFACE  ABOUT  A  POINT 

If  these  equations  be  multiplied  by  — »  —  •>  —  respectively  and 

dx    o-     dz  ^u    ^u    ^u 

added,  and  by  — »  ^ »  —  respectively  and  added,  we  get 

dv    dv    dv 


Fdu  +  Gdv  —  r(D'du  +  D"  dv)  =  0. 
But  these  equations  are  the  same  as  (23).    Hence: 

The  normals  to  a  surface  along  a  curve  of  it  form  a  ruled  surface 
which  is  a  developable  only  when  the  curve  is  a  line  of  curvature  ;  in 
this  case  the  points  of  the  edge  of  regression  are  the  centers  of  normal 
curvature  of  the  surface  in  the  direction  of  the  curve. 

The  coordinates  of  the  principal  centers  of  curvature  are 

(28) 

When  the  parametric  curves  are  the  lines  of  curvature,  equa 
tion  (26)  is  necessarily  of  the  form 

(29)  X  dudv  =  0, 

and  consequently  we  must  have  ED'  —  FD  =  0,  FD"  —  GD'=  0. 
Since  ED"—GD  =£  0,  these  equations  are  equivalent  to 

(30)  ^=0,         D'=Q. 

Conversely,  when  these  conditions  are  satisfied  equation  (26) 
reduces  to  the  form  (29).  Hence: 

A  necessary  and  sufficient  condition  that  the  lines  of  curvature  be 
parametric  is  that  F  and  D'  be  zero. 

Let  the  lines  of  curvature  be  parametric,  and  let  p^  and  p2  denote 
the  principal  radii  of  normal  curvature  of  the  surface  in  the  direc 
tions  of  the  lines  of  curvature  v  =  const,  and  u  =  const,  respectively. 
From  (19)  we  find 

(31)  ^  =  f'         ~  =  ^T 
and  equations  (13)  become 

By  dY      dz  _          dZ 


(32) 


du  du  '     du          ri  du       du 


These  equations  are  called  the  equations  of  Rodrigues. 


TOTAL  AND  MEAN  CURVATURE  123 

52.  Total  and  mean  curvature.  Of  fundamental  importance  in 
the  discussion  of  the  nature  of  a  surface  in  the  neighborhood  of 
a  point  are  the  product  and  the  sum  of  the  principal  curvatures  at 
the  point.  They  are  called  the  total  curvature  *  of  the  surface  at 
the  point  and  the  mean  curvature  respectively.  If  they  be  denoted 
by  K  and  Km1  we  have,  from  (25), 

1 


(33) 


K     Mb"         ^ 
JL-i+i- 


Pi     P*  ^ 

When  K  is  positive  at  a  point  J/,  the  two  principal  radii  have 
the  same  sign,  and  consequently  the  two  centers  of  principal  curva 
ture  lie  on  the  same  side  of  the  tangent  plane.  As  all  the  centers 
of  curvature  of  other  normal  sections  lie  between  these  two,  the 
portion  of  the  surface  in  the  neighborhood  of  M  lies  entirely  on 
one  side  of  the  tangent  plane.  This  can  be  seen  also  in  another 
way.  Since  H2  is  positive,  we  must  have  DDf'  —  D'2  >  0.  Hence 
the  distance  from  a  near-by  point  to  the  tangent  plane  at  Jf,  since 
it  is  proportional  to  the  fundamental  form  <l>  (§  48),  does  not 
change  sign  as  dv/du  is  varied. 

When  K  is  negative  at  M,  the  principal  radii  differ  in  sign,  and 
consequently  part  of  the  surface  lies  on  one  side  of  the  tangent 
plane  and  part  on  the  other.  In  particular  there  are  two  directions, 
given  by  j>du*+2  D'dudv  +  D" dv2  =  0, 

for  which  the  normal  curvature  is  zero.  In  these  directions  the  dis 
tances  of  the  near-by  points  of  the  surface  from  the  tangent  plane, 
as  given  by  (4),  are  quantities  of  the  third  order  at  least.  Hence 
these  lines  are  the  tangents  at  M  to  the  curve  in  which  the  tangent 
plane  at  M  meets  the  surface. 

At  the  points  for  which  K  is  zero,  one  of  the  principal  radii  is 
infinite.  At  these  points  <£  has  the  form  (^/J)  du  -f  ^W'dvf  and 
vanishes  in  the  direction  Vl)du  +  ^/D"dv  =  0.  But  as  dv/du  passes 
through  the  value  given  by  this  equation,  <1>  does  not  change  sign. 
Hence  the  surface  lies  on  one  side  of  the  tangent  plane  and  is  tan 
gent  to  it  along  the  above  direction. 

*  The  total  curvature  is  sometimes  called  the  Gaussian  curvature,  after  the  celebrated 
geometer  who  suggested  it  as  a  suitable  measure  of  the  curvature  at  a  point.  Cf .  Gauss,  p.  15. 


124        GEOMETEY  OF  A  SUEFACE  ABOUT  A  POINT 


An  anchor  ring,  or  tore,  is  a  surface  with  points  of  all  three  kinds.  Such  a  sur 
face  may  be  generated  by  the  rotation  of  a  circle  of  radius  a  about  an  axis  in  the 
plane  of  the  circle  and  at  a  distance  b  (>  a)  from  the  center  of  the  circle.  The 
points  at  the  distance  b  from  the  axis  lie  in  two  circles,  and  the  tangent  plane  to 
the  tore  at  a  point  of  either  of  the  circles  is  tangent  all  along  the  circle.  Hence  the 
surface  has  zero  curvature  at  all  points  of  these  circles.  At  every  point  whose  dis 
tance  from  the  axis  is  greater  than  b  the  surface  lies  on  one  side  of  the  tangent 
plane,  whereas,  when  the  distance  is  less  than  6,  the  tangent  plane  cuts  the  surface. 

There  are  surfaces  for  which  K  is  positive  at  every  point,  as, 
for  example,  the  ellipsoid  and  the  elliptic  paraboloid.  Moreover, 
for  the  hyperboloid  of  one  sheet  and  the  hyperbolic  paraboloid  the 
curvature  is  negative  at  every  point.  Surfaces  of  the  former  type 
are  called  surfaces  of  positive  curvature,  of  the  latter  type  surfaces 
of  negative  curvature. 

Later  (§  64)  we  shall  prove  that  when  K  is  zero  at  all  points  of 
a  surface  the  latter  is  developable,  and  conversely. 

53.  Equation  of  Euler.  Dupin  indicatrix.  When  the  lines  of 
curvature  are  parametric,  equation  (18)  can  be  written,  in  con 
sequence  of  (III,  23)  and  (31),  in  the  form 


(34) 


cos2#      sin26> 
I 

Pi  Pi 


where  0  is  the  angle  between  the  directions  whose  radii  of  normal 
curvature  are  R  and  pr  Equation  (34)  is  called  the  equation  of  Euler. 

When  the  total  curvature  K  at  a  point 
is  positive,  pl  and  p2  for  the  point  have 
the  same  sign,  and  R  has  this  sign  for  all 
directions.  If  the  tangents  to  the  lines 
of  curvature  at  the  point  M  be  taken  for 
coordinate  axes,  with  respect  to  which  % 
FlG-  13  and  T?  are  coordinates,  and  segments  of 

length  ±VTIFi  be  laid  off  from  M  in  the  two  directions  correspond 
ing  to  R,  the  locus  of  the  end  points  of  these  segments  is  the  ellipse 
(fig.  13)  whose  equation  is  & 

N+ra 

This  ellipse  is  called  the  Dupin  indicatrix  for  the  point.  When,  in 
particular,  pl  and  p2  are  equal,  the  indicatrix  is  a  circle.  Hence  the 
Dupin  indicatrix  at  an  umbilical  point  is  a  circle  (§  50).  For  this 
reason  such  a  point  is  sometimes  called  a  circular  point. 


=  1. 


DUPIN  INDICATRIX 


125 


When  K  is  negative  pl  and  p2  differ  in  sign,  and  consequently 
certain  values  of  R  are  positive  and  the  others  are  negative.  In 
the  directions  for  which  R  is  positive  we  lay  off  the  segments 
±  V.Z2,  and  in  the  other  directions  ±  V —  R.  The  locus  of  the 
end  points  of  these  segments  con 
sists  of  the  conjugate  hyperbolas 
(fig.  14)  whose  equations  are 


Pl 


T]_ 
Pt 


We  remark  that  R  is  infinite  for 
the  directions  given  by 

(35)  tah20=-^, 


FIG.  14 


or,  in  other  words,  in  the   directions  of  the   asymptotes   to  the 

hyperbolas.    The  above  locus  is  the  Dupin  indicatrix  for  the  point. 

Finally,  when  K  =  0  the  equation  of  the  indicatrix  is  of  one  of 

the  forms  e-2_  i 


that  is,  a  pair  of  parallel  straight  lines.  In  view  of  the  foregoing 
results,  a  point  of  a  surface  is  called  elliptic,  hyperbolic,  or  parabolic, 
according  as  the  total  curvature  at  the  point  is  positive,  negative, 
or  zero.1 

In  consequence  of  (4)  the  expression  for  the  distance  p  upon  the 
tangent  plane  to  a  surface  at  a  point  M  from  a  near-by  point  P  of 

the  surface  is  given  by 

Edu2      Gdv2      n 
+ =  2  », 


Pl 


to  within  terms  of  higher  order.  But  ^/Edu  and  ^/Gdv  are  the 
distances,  to  within  terms  of  higher  order,  of  P  from  the  normal 
planes  to  the  surface  at  M  in  the  directions  of  the  lines  of  curva 
ture.  Hence  the  plane  parallel  to  the  tangent  plane  and  at  a  dis 
tance  p  from  it  cuts  the  surface  in  the  curve 


Evidently  this  is  a  conic  similar  to  the  Dupin  indicatrix  at  an 
elliptic  or  parabolic  point,  and  to  a  part  of  the  indicatrix  at  a 
hyperbolic  point. 


126        GEOMETRY  OF  A  SURFACE  ABOUT  A  POINT 


EXAMPLES 

1.  Show  that  the  meridians  and  parallels  of  a  surface  of  revolution  are  its  lines 
of  curvature,  and  determine  the  character  of  the  developable  surfaces  formed  by 
the  normals  to  the  surface  along  these  lines. 

2.  Show  that  the  parametric  lines  on  the  surface 

a  ,  b  ,  uv 

X  =  -(tt-M),  y  =  -(U-v),          z  =  -, 

are  straight  lines.    Find  the  lines  of  curvature. 

3.  When  a  surface  is  denned  by  z  =  /(x,  ?/),  the  expressions  for  the  curvatures  are 


and  the  equation  of  the  lines  of  curvature  is 

[(1  +  p2)  s  -  pqr]  (to*  +  [(i  +  p-2)  t-(l  +  g2)  r-j  dxdy  +  [pqt  _  (1  +  ?2)  s]  dyz  =  0. 


4.  The  principal  radii  of  the  surface  y  cos x  sin  -  =  0  at  a  point  (x,  y,  z)  are 

2     i     o  2  _i_  n2 

equal  to  ±  —  —  •    Find  the  lines  of  curvature. 

5.  Derive  the  equations  of  the  tore,  defined  in  §  52,  and  prove  therefrom  the 
results  stated. 

6.  The  sum  of  the  normal  curvatures  in  two  orthogonal  directions  is  constant. 

7.  The  Euler  equation  can  be  written 

E  =  —  2plp* 

Pi  +  P2  -  (PI  -  pa)  cos  2  6 

54.  Conjugate  directions  at  a  point.  Conjugate  systems.  Two 
curves  on  a  surface  through  a  point  M  are  said  to  have  conjugate 
directions  when  their  tangents  at  M  coincide  with  conjugate  diam 
eters  of  the  Dupin  indicatrix  for  the  point.  These  tangents  are 
also  parallel  to  conjugate  diameters  of  the  conicr  in  which  the  sur 
face  is  cut  by  a  plane  parallel  to  the  tangent  plane  to  M  and  very 
near  it.  Let  P  denote  a  point  of  this  conic  and  N  the  point  in 
which  its  plane  a  cuts  the  normal  at  M.  The  tangent  plane  to 
the  surface  at  P  meets  the  plane  a  in  the  tangent  line  at  P  to  the 
conic.  Moreover,  this  tangent  line  is  parallel  to  the  diameter  conju 
gate  to  NP.  Hence  as  P  approaches  3/this  tangent  line  approaches 
the  diameter  of  the  Dupin  indicatrix,  which  is  conjugate  to  the 
diameter  in  the  direction  MP.  Hence  we  have  (cf.  §  27) : 

The  characteristic  of  the  tangent  plane  to  a  surface,  as  the  point 
of  contact  moves  along  a  curve,  is  the  tangent  to  the  surface  in  the 
direction  conjugate  to  the  curve. 


CONJUGATE  DIRECTIONS  127 

By  means  of  this  theorem  we  derive  the  analytical  condition  for 
conjugate  directions. 

If  the  equation  of  the  tangent  plane  is 


f,  77,  f  being  current  coordinates,  the  characteristic  is  denned  by 
this  equation,  and 


where  s  is  the  arc  of  the  curve  along  which  the  point  of  contact 
moves.  If  &c,  8y,  Bz  denote  increments  of  #,  ?/,  z  in  the  direction 
conjugate  to  the  curve,  we  have,  from  the  above  equations, 


If  increments  of  u  and  v  in  the  conjugate  direction  be  denoted  by 
Bu  and  8v,  this  equation  may  be  written 

(36)  D  duBu  +  Df(du8v  +  dvSu)  +  D"dv§v  =  0. 

The  directions  conjugate  to  any  curve  of  the  family 

(37)  <£(%,  v)  =  const. 
are  given  by 

(38) 


cv  du  dv  du 

As  this  is  a  differential  equation  of  the  first  order  and  first  degree, 
it  defines  a  one-parameter  family  of  curves.  These  curves  and  the 
curves  </>  =  const,  are  said  to  form  a  conjugate  system.  Moreover, 
any  two  families  of  curves  are  said  to  form  a  conjugate  system 
when  the  tangents  to  a  curve  of  each  family  at  their  point  of  inter 
section  have  conjugate  directions. 

From  (36)  it  follows  that  the  curves  conjugate  to  the  curves 
v  =  const,  are  defined  by  D  Su  +  D'Sv  =  0.  Consequently,  in  order 
that  they  be  the  curves  u  =  const.,  we  must  have  D1  equal  to  zero. 
As  the  converse  also  is  true,  we  have  : 

A  necessary  and  sufficient  condition  that  the  parametric  curves 
form  a  conjugate  system  is  that  D'  be  zero. 


128        GEO  vlETRY  OF  A  SURFACE  ABOUT  A  POINT 

We  have  seen  (§51)  that  the  lines  of  curvature  are  characterized 
by  the  property  that,  when  they  are  parametric,  the  coefficients  F 
and  D'  are  zero.  Hence  : 

The  lines  of  curvature  form  a  conjugate  system  and  the  only 
orthogonal  conjugate  system. 

If  the  lines  of  curvature  are  parametric,  and  the  angles  which 
a  pair  of  conjugate  directions  make  with  the  tangent  to  the  curve 
v  =  const,  are  denoted  by  0  and  6',  we  have 

a         [G  dv  ar         [GSv 

tan  0  =  ^J—  — ,         tan  6'=  x  -  — , 
MjE <  du  N^  Su 

so  that  equation  (36)  may  be  put  in  the  form 

(39)  tan0tan0'  =  -?H, 

which  is  the  well-known  equation  of  conjugate  directions  of  a  conic. 

55.  Asymptotic  lines.   Characteristic  lines.  When  0'  is  equal  to  0, 

equation  (39)  reduces  to  (35).    Hence  the  asymptotic  directions  are 

self-conjugate.   If  in  equation  (36)  we  put  Sv/Su  =  dv/du,  we  obtain 

(40)  D  du2  +  2  D'dudv  +  D"  dv2  =  0, 

which  determines,  consequently,  the  asymptotic  directions  at  each 
point  of  the  surface.  This  equation  defines  a  double  family  of 
curves  upon  the  surface,  two  of  which  pass  through  each  point 
and  admit  as  tangents  the  asymptotic  directions  at  the  point.  They 
are  called  the  asymptotic  lines  of  the  surface. 

The  asymptotic  lines  are  imaginary  on  surfaces'  of  positive  curva 
ture,  real  on  surfaces  of  negative  curvature,  and  consist  of  a  single 
real  family  on  a  surface  of  zero  curvature. 

Recalling  the  results  of  §  52,  we  say  that  the  tangent  plane  to 
a  surface  at  a  point  cuts  the  surface  in  asymptotic  lines  in  the 
neighborhood  of  the  point.  As  an  immediate  consequence,  we 
have  that  the  generators  of  a  ruled  surface  form  one  family  of 
asymptotic  lines. 

Since  an  asymptotic  line  is  self-conjugate,  the  characteristics  of 
the  tangent  plane  as  the  point  of  contact  moves  along  an  asymp 
totic  line  are  the  tangents  to  the  latter.  Hence  the  osculating 
plane  of  an  asymptotic  line  at  a  point  is  the  tangent  plane  to  the 


ASYMPTOTIC  LINES  129 

surface  at  the  point,  and  consequently  the  asymptotic  line  is  the 
edge  of  regression  of  the  developable  circumscribing  the  surface 
along  the  asymptotic  line.    This  follows  also  from  equation  (16). 
From  (40)  we  have  the  theorem : 

A  necessary  and  sufficient  condition  that  the  asymptotic  lines  upon 
a  surface  be  parametric  is  that 

D=D"=Q. 

If  these  equations  hold,  and,  furthermore,  the  parametric  curves 
are  orthogonal,  it  is  seen  from  (33)  that  the  mean  curvature  is  zero, 
and  conversely.  Hence : 

A  necessary  and  sufficient  condition  that  the  asymptotic  lines  form 
an  orthogonal  system  is  that  the  mean  curvature  of  the  surface  be  zero. 

A  surface  whose  mean  curvature  is  zero  at  every  point  is  called 
a  minimal  surface.  At  each  of  its  points  the  Dupin  indicatrix  con 
sists  of  two  conjugate  equilateral  hyperbolas. 

By  means  of  (39)  we  find  that  the  angle  between  conjugate 
directions  is  given  by 


P«-/>1 

If  we  consider  only  real  lines,  this  angle  can  be  zero  only  for  sur 
faces  of  negative  curvature,  in  which  case  the  directions  are  asymp 
totic.  It  is  natural,  therefore,  to  seek  the  conjugate  directions  upon 
a  surface  of  positive  curvature  for  which  the  included  angle  is  a 
minimum.  To  this  end  we  differentiate  the  right-hand  member  of 
the  above  equation  with  respect  to  6  and  equate  the  result  to  zero. 
The  result  is  reducible  to 
(41)  tan  6  =  : 

Then  from  (39)  we  have 


From  these  equations  it  follows  that  #'  =  —  0,  and 


Conversely,  when  0'  =  —  6  equation  (39)  becomes  (41).    Hence: 

Upon  a  surface  of  positive  curvature  there  is  a  unique  conjugate 
system  for  which  the  angle  between  the  directions  at  any  point  is  the 


130        GEOMETRY  OF  A  SURFACE  ABOUT  A  POINT 

minimum  angle  between  conjugate  directions  at  the  point ;  it  is  the 
only  conjugate  system  whose  directions  are  symmetric  with  respect 
to  the  directions  of  the  lines  of  curvature. 

These  lines  are  called  the  characteristic  lines.  It  is  of  interest 
to  note  that  equations  (35)  and  (41)  are  similar,  and  that  the  real 
asymptotic  directions  upon  a  surface  of  negative  curvature  are 
symmetric  with  respect  to  the  directions  of  the  lines  of  curvature. 

As  just  seen,  if  6  is  the  angle  which  one  characteristic  line  makes 
with  the  line  of  curvature  v  =  const,  at  a  point,  the  other  charac 
teristic  line  makes  the  angle  —  6.  Hence  the  radii  of  normal  curva 
ture  for  these  directions  are  equal,  and  consequently  a  necessary 
and  sufficient  condition  that  the  characteristic  curves  of  a  surface 
be  parametric  is 

(42)  f  =  7T         iy=°' 

56.  Corresponding  systems  on  two  surfaces.  By  reasoning  similar 
to  that  of  §  34  we  establish  the  theorem : 

A  necessary  and  sufficient  condition  that  the  curves  defined  by 
fidu2+2S dudv  +  T dv2  =  0  form  a  conjugate  system  upon  a  sur- 

face  *s  RD"  +  TD  —  2  SD'  =  0. 

From  this  we  have  at  once : 

If  the  second  quadratic  forms  of  two  surfaces  S  and  Sl  are 
D  du2  +  2  D' dudv  +  D"  dv2  and  Dl  du2  +  2  D[  dudv  +  D[f  dv\  and  if  a 
point  on  one  surface  is  said  to  correspond  to  the  point  on  the  other 
with  the  same  values  of  u  and  tf,  the  equation 

du2        D?    D" 

(43)  —dudv     D[     D' 

dv2        DI     D 

defines  a  system  of  curves  which  is  conjugate  for  both  surfaces. 

By  the  methods  of  §  50  we  prove  that  these  curves  are  real  when 
either  or  both  of  the  surfaces  S,  Sl  is  of  positive  curvature.  If  the 
curvature  of  S  is  negative  and  it  is  referred  to  its  asymptotic  lines, 
the  above  equation  reduces  to 


GEODESIC  CURVATUKE  131 

Hence  the  system  is  real  when  Dl  and  D['  have  the  same  sign,  that 
is,  when  the  curvature  of  Sl  is  positive. 

Another  consequence  of  the  above  theorem  is  : 

A  necessary  and  sufficient  condition  that  asymptotic  lines  on  one  of 
two  surfaces  $,  Sl  correspond  to  a  conjugate  system  on  the  other  is 

(44)  DD'J  +  D"Di  -  2  D'D[  =  0. 


EXAMPLES 

1.  Find  the  curves  on  the  general  surface  of  revolution  which  are  conjugate  to 
the  loxodromic  curves  which  cut  the  meridians  under  the  angle  a. 

2.  Find  the  curves  on  the  general  right  conoid,  Ex.  1,  p.  56,  which  are  conju 
gate  to  the  orthogonal  trajectories  of  the  generators. 

3.  When  the  equations  of  a  surface  are  of  the  form 

x=Ul,        y=Vi,        z=U2+V2, 

where  U\  and  U2  are  functions  of  u  alone,  and  V\  and  F2  of  v  alone,  the  para 
metric  curves  are  plane  and  form  a  conjugate  system. 

4.  Prove  that  the  sum  of  normal  radii  at  a  point  in  conjugate  directions  is 
constant. 

5.  When  a  surface  of  revolution  is  referred  to  its  meridians  and  parallels,  the 
asymptotic  lines  can  be  found  by  quadratures. 

6.  Find  the  asymptotic  lines  on  the  surface 

acosw 
x  =  a(l  +  cos  it)  cot  v,        y  =  a(l  +  cosw),        z=  —  -- 

7.  Determine  the  asymptotic  lines  upon  the  surface  z  —y  sin  a:  and  their  orthog 
onal  trajectories.    Show  that  the  x-axis  belongs  to  one  of  the  latter  families. 

8.  Find  the  asymptotic  lines  on  the  surface  2  ?/3  -  2  xyz  +  z2  =  0,  and  determine 
their  projections  on  the  xy-plane. 

9.  Prove  that  the  product  of  the  normal  radii  in  conjugate  directions  is  a  maxi 
mum  for  characteristic  lines  and  a  minimum  for  lines  of  curvature. 

10.  When  the  parametric  lines  are  any  whatever,  the  equation  of  character 
istic  lines  is 

[D(GD  -  ED")  -  2D'(FD  -  ED')]  tin*  +  2  [D'(GD  +  ED")  -  2  FDD"]  dudv 
+  [2D'(GD'  -  FD")  -  D"(GD  -  ED")]  dv*  =  0. 

57.  Geodesic  curvature.  Geodesies.  Consider  a  curve  C  upon  a 
surface  and  the  tangent  plane  to  the  surface  at  a  point  M  of  C. 
Project  orthogonally  upon  this  tangent  plane  the  portion  of  the 
curve  in  the  neighborhood  of  M,  and  let  C1  denote  this  projection. 
The  curve  C1  is  a  normal  section  of  the  projecting  cylinder,  and  C 
is  a  curve  upon  the  latter,  tangent  to  C"  at  M.  Hence  the  theorem 


132        GEOMETRY  OF  A  SURFACE  ABOUT  A  POINT 

of  Meusnier  can  be  applied  to  these  two  curves.  If  l/pg  denotes  the 
curvature  of  C'  and  -^  the  angle  between  the  principal  normal  to  C 
and  the  positive  direction  of  the  normal  to  the  cylinder  at  Jf,  we  have 

(45)  i  =  c^. 

P,         P 

In  order  to  connect  this  result  with  others,  it  is  necessary  to 
define  the  positive  direction  of  the  normal  to  the  cylinder.  This 
normal  lies  in  the  tangent  plane  to  the  surface.  We  make  the 
convention  that  the  positive  directions  of  the  tangent  to  the  curve, 
the  normal  to  the  cylinder,  and  the  normal  to  the  surface  shall 
have  the  same  mutual  orientations  as  the  positive  or-,  y-,  and  2-axes. 
From  this  choice  of  direction  it  follows  that  if,  as  usual,  the  direc 
tion-cosines  of  the  tangent  to  the  curve  be  dx/ds,  dy/ds,  dz/ds,  then 
those  of  the  normal  to  the  cylinder  are 

(46)  Y*-Z*/,          Z~-*~>          *f-4- 

ds         ds  ds         ds  ds          ds 

The  curvature  of  C f  is  called  the  geodesic  curvature  of  (7,  and  pg 
the  radius  of  geodesic  curvature.  And  the  center  of  curvature  of  C' 
is  called  the  center  of  geodesic  curvature  of  C. 

From  its  definition  the  geodesic  curvature  is  positive  or  nega 
tive  according  as  the  osculating  plane  of  C  lies  on  one  side  or  the 
other  of  the  normal  plane  to  the  surface  through  the  tangent  to  C. 
From  (45)  it  follows  that  the  center  of  first  curvature  of  C  is  the 
projection  upon  its  osculating  plane  of  the  center  of  geodesic 
curvature.  Moreover,  the  former  is  also  the  projection  of  the 
center  of  curvature  of  the  normal  section  tangent  to  C  (§49). 
Hence  the  plane  through  a  point  M  of  (7,  normal  to  the  line 
joining  the  centers  of  normal  and  geodesic  curvature  at  M,  is  the 
osculating  plane  of  C  for  this  point,  and  its  intersection  with  the 
join  is  the  center  of  first  curvature. 

By  definition  (§  49)  w  denotes  the  angle  which  the  positive 
direction  of  the  normal  to  the  surface  makes  with  the  positive 
direction  of  the  principal  normal  to  (7,  angles  being  measured 
toward  the  binomial.  Hence  equation  (45)  can  be  written 

1      sin  w 

(47)  -  = 


GEODESIC  CUEVATUKE 


133 


These  various  quantities  are  represented  in  fig.  15,  for  which 
the  tangent  to  the  curve  is  normal  to  the  plane  of  the  paper,  and 
is  directed  toward  the  reader.  The  directed  lines  MP,  MB,  MK, 
MN  represent  respectively  the  positive  directions  of  the  principal 
normal  and  binomial  of  the  curve  and 
the  normals  to  the  projecting  cylinder 
and  to  the  surface. 

A  curve  whose  principal  normal  at 
every  point  coincides  with  the  normal 
to  the  surface  upon  which  it  lies,  is 
called  a  geodesic.  From  (45)  it  follows 
that  a  geodesic  may  also  be  defined  as 
a  curve  whose  geodesic  curvature  is 
zero  at  every  point.  For  example,  the 
meridians  of  a  surface  of  revolution  are 
geodesies,  as  follows  from  the  results  in  §  46.  A  twisted  curve  is  a 
geodesic  on  its  rectifying  developable,  and  when  a  straight  line 
lies  011  a  surface,  it  is  a  geodesic  for  the  surface.  Later  we 
shall  make  an  extensive  study  of  geodesies,  but  now  we  desire 
to  find  an  expression  for  the  geodesic  curvature  in  terms  of 
the  fundamental  quantities  of  the  surface  and  the  equation  of 
the  curve. 

58.  Fundamental  formulas.    The  direction-cosines  of  the  prin 
cipal  normal  are  (§  8) 

d2x       ~    d2y  d2z 

f\  _ —   «  Q  •  Q  • 

H  ds2         ^  ds2         r  ds2     . 

Consequently,  by  means   of  (46),   equation   (45)   may  be   put  in 

the  form 

1      ^\  /     dz         dy\  d  x 

(48)  7g=*\     ds~     ~ds)~ds2' 

Expressed  as  functions  of  u  and  v,  the  quantities  -j-'  -ji  are  °^ 


the  form 


dx  _  fa  du      dx  dv 
ds      du  ds      dv  ds 


_  g  d2x   dudv      d*x  ^Y+  —  —  +  —  — 

~ds2~'du2  \ds)         dudv  ds  ds      dv2  \ds)      du  ds2      dv  ds2 


134        GEOMETRY  OF  A  SURFACE  ABOUT  A  POINT 

When  these  expressions  are  substituted  in  (48),  and  in  the  reduction 
we  make  use  of  (10)  and  (11),  we  obtain 


ds          ds 
where  L  and  M  have  the  significance 


_                                 ,_                       ,  F  (Fv 

~+       ds  ds^\dv     29»/Vb/        <*r  ds^ 

--         .     —\\  ^^  —  +  ~  —  (—\\  F—  G- 
~    du  ~  2  ~dv)  \ds)      du  ds  ds      2  dv  \ds)          ds* 


* 


From  this  it  is  seen  that  the  geodesic  curvature  of  a  curve  depends 
upon  E,  F,  G,  and  is  entirely  independent  of  D,  D',  D". 

Suppose  that  the  parametric  lines  form  an  orthogonal  system, 
and  that  the  radius  of  geodesic  curvature  of  a  curve  v  =  const,  be 
denoted  by  pgu.  In  this  case  F=  0,  ds  =  Vfldu.  Hence  the  above 
equation  reduces  to 

(50)  r— - 


In  like  manner  we  find  that  the  geodesic  curvature  of  a  curve 
u  =  const,  is  given  by 

_! 


As  an  immediate  consequence  c^  these  equations  we  have  the 
theorem :  f 

When  the  parametric  lines  upon  a  surface  form  an  orthogonal 
system,  a  necessary  and  sufficient  condition  that  the  curves  v  =  const, 
or  u  =  const,  be  geodesies  is  that  E  be  a  function  of  u  alone  or  G  of  v 
alone  respectively. 

It  will  now  be  shown  that  pgu  is  expressible  as  a  function  of 
differential  parameters  of  v  formed  with  respect  to  the  linear  ele 
ment  (III,  4). 

From  the  definition  of  these  parameters  (§§  37,  38)  it  follows 
that  when  ^=0 

l     d    IE 

~V\G' 


GEODESIC  CURVATURE  135 

Hence,  by  substitution  in  (50),  we  obtain 

(52)  JUPAL      ./        1 
Pgn         [y&^v 

In  like  manner,  we  find 

(53)  -i-  = 


Thus  we  have  shown  that  the  geodesic  curvature  of  a  parametric 
line  is  a  differential  parameter  of  the  curvilinear  coordinate  of  the 
line.  Since  this  curvature  is  a  geometrical  property  of  a  line,  it  is 
necessarily  independent  of  the  choice  of  parameters,  and  thus  is 
an  invariant.  This  was  evident  a  priori,  but  we  have  just  shown 
that  it  is  an  invariant  of  the  differential  parameter  type. 

From  the  definition  of  the  positive  direction  of  the  normal  to  a 
surface  (§  48),  and  the  normal  to  the  cylinder  of  projection,  it  fol 
lows  that  the  latter  for  a  curve  v  =  const,  is  the  direction  in  which  v 
increases,  whereas,  for  a  curve  u  —  const.,  it  is  the  direction  in  which 
u  decreases.  Hence,  if  the  latter  curves  be  defined  by  — u  =  const., 
equations  (52)  and  (53)  have  the  same  sign. 

If,  now,  we  imagine  the  surface  referred  to  another  parametric 
system,  for  which  the  linear  element  is 

•  (54)  ds*  =  Edu2  +  2  Fdudv  +  G  dv\ 

the  curve  whose  geodesic  curvature  is  given  by  (50)  will  be  defined 
by  an  equation  such  as  c/>  (u,  v)  =  const.  And  if  the  sign  of  $  be 
such  that  <£  is  increasing  in  the  direction  of  the  normal  of  its  pro 
jecting  cylinder,  its  geodesic  curvature  will  be  given  by 

(55) 

p. 

where  the  differential  parameters  are  formed  with  respect  to  (54). 
If  two  surfaces  are  applicable,  and  points  on  each  with  the  same 
curvilinear  coordinates  correspond,  the  geodesic  curvature  of  the 
curve  <£=  const,  on  each  at  corresponding  points  will  be  the  same 
in  consequence  of  (55).  Hence  : 

Upon  two  applicable  surfaces  the  geodesic  curvature  of  corresponding 
curves,  at  corresponding  points,  is  the  same. 


136        GEOMETRY  OF  A  SURFACE  ABOUT  A  POINT 

When   the   second  member  of  equation  (55)  is  developed  by 
(III,  46,  56),  we  have 


1        1 


ff(V*d 


d  \      du          dv\       d  \      dv          du 


cu 


H 

L    1 

cu 


dv 


R 


dv          du    d        1 


H 


FW\ 
I  I  d  I      du          dv\       d^\      dv          du, 


Hence  we  have  the  formula  of  Bonnet*: 


(56)        i-^ 


du          dv 


In  particular,  the  geodesic  curvature  of  the  parametric  curves, 
when  the  latter  do  not  form  an  orthogonal  system,  is  given  by 


(57) 


ft.    « 


dv 


du 


The  geodesic  curvature  of  a  curve  of  the  family,  defined  by 

the  differential  equation 

Mdu  +  Ndv*sQ, 

has  the  value 

1  _    1   f  d  I FN—  GM  \ 

pg  "  H  \  du  \^EN*-  2  FMN+  GM2/ 
d  /  FM-EN 


ZFMN+  GM 


0 


*Memoire  sur  la  theorie  generale  des  surfaces,  Journal  de  VEcole  Poll/technique, 
Cahier  32  (1848),  p.  1. 


GEODESIC  TOKSION  137 

In  illustration  of  the  preceding  results,  we  establish  the  theorem  : 

When  the  curves  of  an  orthogonal  system  have  constant  geodesic  curvature,  the 
system  is  isothermal. 

When  the  surface  is  referred  to  these  lines,  and  the  linear  element  is  written 
dsz  =  Edu2  +  Gdv2,  the  condition  that  the  geodesic  curvature  of  these  curves  be 
constant  is,  by  (50)  and  (51), 

1      dVE 
l/i,  —  -  —  =  KI, 

VEG   fa 

where  Ui  and  V\  are  functions  of  u  and  v  respectively.    If  these  equations  are 
differentiated  with  respect  to  D  and  u  respectively,  we  get 


dudv  dv  cu  cucv  dv  du 

""S  75^ 

Subtracting,  we  obtain  -  log—  =  0. 

CUCV  G 

Hence  E/G  is  equal  to  the  ratio  of  a  function  of  u  and  a  function  of  u,  and  the 
system  is  isothermic.  In  terms  of  isothermic  parameters,  equations  (i)  are  of 
the  form 

10X  =  rr,  J.3X 

\2  5U  \2  dV  ~ 

and  the  linear  element  is 

(II)  «tf 

It  is  evident  that  the  meridians  and  parallels  on  a  surface  of  revolution  form 
such  a  system.  The  same  is  true  likewise  of  an  orthogonal  system  of  small  circles 
on  a  sphere. 

59.  Geodesic  torsion.  We  have  just  seen  that  when  a  curve  is 
denned  by  a  finite  equation  or  a  differential  equation,  its  geodesic 
curvature  can  be  found  directly.  The  same  is  true  of  the  normal 
curvature  of  the  surface  in  the  direction  of  the  curve  by  (18). 
Then  from  (16)  and  (47)  follow  the  expressions  for  p  and  tw.  In 
order  to  define  the  curve  it  remains  for  us  to  obtain  an  expression 
for  the  torsion. 

From  the  definition  of  o>  it  follows  that 

(59)  sin  o)  =  X\  -f  Y/JL  +  Zv, 

where  X,  /Lt,  v  are  the  direction-cosines  of  the  binormal.  If  this 
equation  is  differentiated  with  respect  to  the  arc  of  the  curve, 
and  the  Frenet  formulas  (I,  50)  are  used  in  the  reduction,  we  get 


(60)  v  ^       _i      *»-  ds 


138        GEOMETRY  OF  A  SURFACE  ABOUT  A  POINT 

From  (I,  37,  41)  Ave  have 

dx  d"*z\ 
~d~s~dfj 


_     /dy  d"z      dz  d*y\  _    /dz  d*x  _  dx  d"*z 

~~~  ^~  ~'~~~~ 


j  ds2      ds  ds2 
rd2x     -V-K  „, 


and  r-  /  .—   7  . 

6?S^ 

Moreover,  from  (13),  we  obtain  the  identity 

ds  ds       ds  ds      H\_  \ds/  ds  ds 


Consequently  equation  (60)  is  equivalent  to 

(61)  eos 

where  l/T  has  the  value 

1       (FD -  ED'}  du2  +(GD—  ED")  dudv  +  ( GD' -  FD")  dv* 
'   "'   ~T  ~  H(Edu2  +  2  Fdudv  +  G  dv2) 

When  cos  w  is  different  from  zero,  that  is,  when  the  curve  is  not 
an  asymptotic  line,  equation  (61)  becomes 

hf-F 

As  the  expression  for  T  involves  only  the  fundamental  coeffi 
cients  and  dv/du,  we  have  the  following  theorem  of  Bonnet: 

The  function —  is  the  same  for  all  curves  which  have  the  same 

T       ds 

tangent  at  a  common  point. 

Among  these  curves  there  is  one  geodesic,  and  only  one,  for  it 
will  be  shown  later  (§  85)  that  one  geodesic  and  only  one  passes 
through  a  given  point  and  has  a  given  direction  at  the  point. 
At  every  point  of  this  geodesic  w  is  equal  to  0°  or  180°,  and  conse 
quently  T—  T.  Hence  the  value  of  T  for  a  given  point  and  direc 
tion  is  that  of  the  radius  of  torsion  of  the  geodesic  with  this  direction. 
The  function  T  is  therefore  called  the  radius  of  geodesic  torsion  of 


GEODESIC  TORSION  139 

the  curve.  From  (63)  it  is  seen  that  T  is  the  radius  of  torsion  of 
any  curve  whose  osculating  plane  makes  a  constant  angle  with  the 
tangent  plane.* 

When  the  numerator  of  the  right-hand  member  of  equation  (62) 
is  equated  to  zero,  we  have  the  differential  equation  of  lines  of 
curvature.  Hence  : 

A  necessary  and  sufficient  condition  that  the  geodesic  torsion  of 
a  curve  be  zero  at  a  point  is  that  the  curve  be  tangent  to  a  line  of 
curvature  at  the  point. 

The  geodesic  torsion  of  the  parametric  lines  is  given  by 
1  _  FD-ED'          l_  _  GD'-FD" 
~T~        EH  ~TV~         GH 

When  these  lines  form  an  orthogonal  system  Tu  and  Tv  differ  only 
in  sign.  Consequently  the  geodesic  torsion  at  the  point  of  meeting 
of  two  curves  cutting  orthogonally  is  the  same  to  within  the  sign. 
Thus  far  in  the  consideration  of  equation  (61)  we  have  excluded 
the  case  of  asymptotic  lines.  In  considering  them  now,  we  assume 
that  they  are  parametric.  The  direction-cosines  of  the  tangent  and 
binomial  to  a  curve  v  =  const,  in  this  case  are 

JL^,         £=-L^,         7=  —  —  ; 

~  ' 


where  e  is  +1  or  —  1.    Consequently  the  direction-cosines  of  the 
principal  normal  have  the  values 


and  similar  expressions  for  m  and  n. 
When  in  the  Frenet  formulas 

d\      I  dfju  _m          dv  _n 

ds  ~~  T          ds       T  ds      r 

we  substitute  the  above  values,  and  in  the  reduction  make  use 
of  (11)  and  (13),  we  get 

(65) 

*  Thus  far  exception  must  be  made  of  asymptotic  lines,  but  later  this  restriction  will 
be  removed. 


140       GEOMETRY  OF  A  SURFACE  ABOUT  A  POINT 

In  like  manner,  the  torsion  of  the  asymptotic  lines  u  =  const,  is 
found  to  be  V—  K.  But  from  (64)  we  find  that  the  geodesic  torsiqn 
in  the  direction  of  the  asymptotic  lines  is  qp  V— JT.  Hence  equation 
(63)  is  true  for  the  asymptotic  lines  as  well  as  for  all  other  curves 
on  the  surface. 

Incidentally  we  have  established  the  following  theorem  of  Enneper : 

The  square  of  the  torsion  of  a  real  asymptotic  line  at  a  point  is  equal  to  the  abso 
lute  value  of  the  total  curvature  of  the  surface  at  the  point;  the  radii  of  torsion  of  the 
asymptotic  lines  through  a  point  differ  only  in  sign. 

The  following  theorem  of  Joachimsthal  is  an  immediate  consequence  of  (63) : 

When  two  surfaces  meet  under  a  constant  angle,  the  line  of  intersection  is  a  line 
of  curvature  of  both  or  neither;  and  conversely,  when  the  curve  of  intersection  of  two 
surfaces  is.  a  line  of  curvature  of  both  they  meet  under  constant  angle. 

For,  if  &>!,  o>2  denote  the  values  of  w  for  the  two  surfaces,  and  Z\,  T2  the  values 
of  T,  we  have,  by  subtracting  the  two  equations  of  the  form  (63),  that  TI  =  T2, 
which  proves  the  first  part  of  the  theorem.  Conversely,  if  \/T\  =  l/T2  =  0,  we 

have  —  (wi  —  uz)  =  0,  and  consequently  the  surfaces  meet  under  constant  angle. 
ds 

EXAMPLES 

1.  Show  that  the  radius  of  geodesic  curvature  of  a  parallel  on  a  surface  of 
revolution  is  the  same  at  all  points  of  the  parallel,  and  determine  its  geometrical 
significance. 

2.  Find  the  geodesic  curvature  of  the  parametric  lines  on  the  surface 

a ,  b  /  uv 

X  =  -(u  +  v),         y  =  -(u-v),          z  =  —> 

3.  Given  a  family  of  loxodromic  curves  upon  a  surface  of  revolution  which  cut 
the  meridians  under  the  same  angle  a ;  show  that  the  geodesic  curvature  of  all  these 
curves  is  the  same  at  their  points  of  intersection  with  a  parallel. 

4.  Straight  lines  on  a  surface  are  the  only  asymptotic  lines  which  are  geodesies. 

5.  Show  that  the  geodesic  torsion  of  a  curve  is  given  by 

1       1/1       1\  . 

—  =  -I )  sin  2  0, 

T      *\fi      pj 

where  6  denotes  the  angle  which  the  direction  of  the  curve  at  a  point  makes  with 
the  line  of  curvature  v  =  const,  through  the  point. 

6.  Every  geodesic  line  of  curvature  is  a  plane  curve. 

7.  Every  plane  geodesic  line  is  a  line  of  curvature. 

8.  When  a  surface  is  cut  by  a  plane  or  a  sphere  under  constant  angle,  it  is  a  line 
of  curvature  on  the  surface,  and  conversely. 

9.  If  the  curves  of  one  family  of  an  isothermal  orthogonal  system  have  constant 
geodesic  curvature,  the  curves  of  the  other  family  have  the  same  property. 


SPHERICAL  REPRESENTATION 


141 


60.  Spherical  representation.  In  the  discussion  of  certain  prop 
erties  of  a  surface  S  it  is  of  advantage  to  make  a  representation  of 
S  upon  the  unit  sphere  *  by  drawing  radii  of  the  sphere  parallel  to 
the  positive  directions  of  the  normals  to  S,  and  taking  the  extrem 
ities  of  the  radii  as  spherical  images  of  the  corresponding  points  on 
S.  As  a  point  M  moves  along  a  curve  on  $,  its  image  m  describes 
a  curve  on  the  sphere.  If  we  limit  our  consideration  to  a  portion 
of  the  surface  in  which  no  two  normals  are  parallel,  the  portions 
of  the  surface  and  sphere  will  be  in  a  one-to-one  correspondence. 
This  map  of  the  surface  upon  the  sphere  is  called  the  spherical 
representation  of  the  surface,  or  the  Gf-aussian  representation.  It 
was  first  employed  by  Gauss  in  his  treatment  of  the  curvature  of 
surfaces.  f 

The  coordinates  of  m  are  the  direction-cosines  of  the  normal  to 
the  surface,  namely  X,  F,  Z,  so  that  if  we  put 


the  square  of  the  linear  element  of  the  spherical  representation  is 

In  §  48  we  established  the  following  equations : 

_f      FD1—  GD  dx  .  FD  —  ED'  dx 
%» 
(68) 


du~         H2 du  H2 dv 

dX      FD"—GD'  ex      FD'  —  ED"  dx 


]v  Hz         du  H2         dv 

By  means  of  these  relations  and  similar  ones  in  F  and  Z,  the  func 
tions  (o,  c^,  &  may  be  given  the  forms 


(69) 


=  ~  [GIf  -  2 
H 


_ 

JL 
H 


F(DD"  +  D'2)  +  ED'D"], 


-  2 


or,  in  terms  of  the  total  and  mean  curvatures  (§  52), 
(70)       £ 


*  The  sphere  of  unit  radius  and  center  at  the  origin  of  coordinates.        t  L.c.,  p.  9. 


142       GEOMETRY  OF  A  SUEFACE  ABOUT  A  POINT 

In  consequence  of  these  relations  the  linear  element  (67)  may  be 
given  the  form 


(71)  do-z—Itm(Ddu-+  2D'dudv+D"dv2)— K(Edi?  +  2  Fdudv 
and,  by  (18), 

(72)  ^=(§ 

\  J* 

From  (70)  we  have  also 

(73)  ff. 


where  e  is  ±1,  according  as  K  is  positive  or  negative. 

Equations  (69)  are  linear  in  E,  F,  G.    Solving  for  the  latter, 
we  have 

E  =  — 


(74) 


In  seeking  the  differential  equation  of  the  lines  of  curvature 
from  the  definition  that  the  normals  to  the  surface  along  such  a 
curve  form  a  developable  surface,  we  found  (§  51)  that  for  a  dis 
placement  in  the  direction  of  a  line  of  curvature  we  have 

fa      7  &£     7  l^X    3  ,      2X     ,     \  . 

—  du  +  —  dv  +  r(  —  du  +  — -  dv    =  0, 

du  dv  \du  dv      / 

and  similar  equations  in  y  and  2,  where  r  denotes  the  radius  of 
principal  curvature  for  the  direction.    If  these  equations  be  multi- 

o  -T7"       QT/"        Q  *7  7}  "jf 

plied  respectively  by  -— »  -—  •>  --  and  added,  and  likewise  by  -— > 


du     du     du  dv 

and  added,  th 

dv     dv 


?±.,  _  an(l  added,  the  resulting  equations  may  be  written 


D  du  +D'dv  -  r(fdu+£di>)  =  0, 
D'du  +Dndv  -r(3du  +  gdv}  =  0. 

Eliminating  r,  we  have  as  the  equation  of  the  lines  of  curvature 

(75)  -     '2  -"dudv       D'-D"&dv*  =  0. 


SPHERICAL  REPRESENTATION  143 

Again,  the  elimination  of  du  and  dv  gives  the  equation  of  the 
principal  radii  in  the  form 

(76)     (<o£—  c^2) r*—(<oD" +  3D  —  2 &D')  r  +  (DD"  —  D' )  =  0, 

so  that 

(77) 


These  results  enable  us  to  write  equations  (74)  thus  : 
(78) 


61.  Relations  between  a  surface  and  its  spherical  representation. 
Since  the  radius  of  normal  curvature  R  is  a  function  of  the  direc 
tion  except  when  the  surface  is  a  sphere,  we  obtain  from  (72)  the 
following  theorem  : 

A  necessary  and  sufficient  condition  that  the  spherical  representation 
of  a  surface  be  conformal  is  that  the  surface  be  minimal  or  a  sphere. 

As  a  consequence  of  this  theorem  we  have  that  every  orthog 
onal  system  on  a  minimal  surface  is  represented  on  the  sphere  by 
an  orthogonal  system.  From  (70)  it  is  seen  that  if  a  surface  is 
not  minimal,  the  parametric  systems  on  both  the  surface  and  the 
sphere  can  be  orthogonal  only  when  If  is  zero,  that  is,  when  the 
lines  of  curvature  are  parametric.  Hence  we  have  : 

The  lines  of  curvature  of  a  surface  are  represented  on  the  sphere 
by  an  orthogonal  system  ;  this  is  a  characteristic  property  of  lines 
of  curvature,  unless  the  surface  be  minimal. 

This  theorem  follows  also  as  a  direct  consequence  of  the  theorem  : 

A  necessary  and  sufficient  condition  that  the  tangents  to  a  curve 
upon  a  surface  and  to  its  image  at  corresponding  points  be  parallel  is 
that  the  curve  be  a  line  of  curvature. 

In  order  to  prove  this  theorem  we  assume  that  the  curve  is 
parametric,  v  =  const.  Then  the  condition  of  parallelism  is 

&£.£?.££-  to  •;•&.* 

du     du    du      du    du    du 


144        GEOMETRY  OF  A  SURFACE  ABOUT  A  POINT 

From  (68)  it  follows  that  in  this  case  (FD  —  ED1)  must  be  zero. 
But  the  latter  is  the  condition  that  the  curves  v  =  const,  be  lines  of 
curvature  (§  51).  Moreover,  from  (32)  it  follows  that  the  positive 
half-tangents  to  a  line  of  curvature  and  its  spherical  representa 
tion  have  the  same  or  contrary  sense  according  as  the  correspond 
ing  radius  of  normal  curvature  is  negative  or  positive. 

In  consequence  of  (7)  the  equation  (40)  of  the  asymptotic 
directions  may  be  written 

dxdX+  dydY+  dzdZ  =  0. 
And  so  we  have  the  theorem : 

The  tangents  to  an  asymptotic  line  and  to  its  spherical  representa 
tion  at  corresponding  points  are  perpendicular  to  one  another ;  this 
property  is  characteristic  of  asymptotic  lines. 

It  is  evident  that  the  direction-cosines  of  the  normal  to  the 
sphere  are  equal  to  X,  Y,  Z,  to  within  sign  at  most.  Let  them 
be  denoted  by  X><>  I/,  ^;  then 

(79)  x>  =  —  (—  —  -  —  ? 

#\du  dv       du  dv 

When  expressions  similar  to  (68)  are  substituted  for  the  quantities 
in  the  parentheses,  the  latter  expression  is  reducible  to  KHX. 
Hence,  in  consequence  of  (73),  we  have 

(80)  x>  =  ex,       ^  =  er,       l  =  *z, 

where  e  =  ±  1  according  as  the  curvature  of  the  surface  is  positive 
or  negative. 

From  the  above  it  follows  that  according  as  a  point  of  a  surface 
is  elliptic  or  hyperbolic  the  positive  sides  of  the  tangent  planes  at 
corresponding  points  of  the  surface  and  the  sphere  are  the  same  or 
different.  Suppose,  for  the  moment,  that  the  lines  of  curvature  are 
parametric.  From  our  convention  about  the  positive  direction  of 
the  normal  to  a  surface,  and  the  above  results,  it  follows  that  both 
the  tangents  to  the  parametric  curves  through  a  point  M  have  the 
same  sense  as  the  corresponding  tangents  to  the  sphere,  or  both 
have  the  opposite  sense,  when  M  is  an  elliptic  point;  but  that 
one  tangent  has  the  same  sense  as  the  corresponding  tangent  to 
the  sphere,  and  the  other  the  opposite  sense,  when  the  point  is 


GAUSSIAN  CURVATURE  145 

hyperbolic.  Hence,  when  a  point  describes  a  closed  curve  on  a 
surface  its  image  describes  a  closed  curve  on  the  sphere  in  the 
same  or  opposite  sense  according  as  the  surface  has  positive  or 
negative  curvature.  We  say  that  the  areas  inclosed  by  these 
curves  have  the  same  or  opposite  signs  in  these  respective  cases. 

Suppose  now  that  we  consider  a  small  parallelogram  on  the  sur 
face,  whose  vertices  are  the  points  (u,  v),  (u  -f  du,  v),  (u,  v  -f  dv), 
and  (u  +  du,  v  +  dv).  The  vertices  of  the  corresponding  parallelo 
gram  on  the  sphere  have  the  same  curvilinear  coordinates,  and 
the  areas  are  Ifdudv  and  e/tdudv,  where  e  ±1  according  as  the  sur 
face  has  positive  or  negative  curvature  in  the  neighborhood  of  the 
point  (u,  v).  The  limiting  value  of  the  ratio  of  the  spherical  and 
the  surface  areas  as  the  vertices  of  the  latter  approach  the  point 
(u,  v)  is  a  measure  of  the  curvature  of  the  surface  similar  to  that 
of  a  plane  curve.  In  consequence  of  (73)  this  limiting  value  is  the 
Gaussian  curvature  K.  Since  any  closed  area  may  be  looked  upon 
as  made  up  of  such  small  parallelograms,  we  have  the  following 
theorem  of  Gauss : 

The  limit  of  the  ratio  of  the  area  of  a  closed  portion  of  a  surface  to 
the  area  of  the  spherical  image  of  it,  as  the  former  converges  to  a  point, 
is  equal  in  value  to  the  product  of  the  principal  radii  at  the  point. 

Since  the  normals  to  a  developable  surface  along  a  generator  are 
parallel,  there  can  be  no  closed  area  for  which  there  are  not  two  nor 
mals  which  are  parallel.  Hence  spherical  representation,  as  defined 
in  §  60,  applies  only  to  nondevelopable  surfaces,  but  so  far  as  the 
preceding  theorem  goes,  it  is  not  necessary  to  make  this  exception ; 
for  the  total  curvature  of  a  developable  surface  is  zero  (§  64), 
and  the  area  of  the  spherical  image  of  any  closed  area  on  such  a 
surface  is  zero. 

The  fact  that  the  Gaussian  curvature  is  zero  at  all  points  of  a  developable  surface, 
whereas  such  a  surface  is  surely  curved,  makes  this  measure  not  altogether  satis 
factory,  and  so  others  have  been  suggested.  Thus,  Sophie  Germain*  advocated 

the  mean  curvature,  and  Casorati  f  has  put  forward  the  expression  -  [ 1 )  • 

2  \Pi       fill 

But  according  to  the  first,  the  curvature  of  a  minimal  surface  is  zero,  and  according 
to  the  second,  a  minimal  surface  has  the  same  curvature  as  a  sphere.  Hence  the 
Gaussian  curvature  continues  to  be  the  one  most  frequently  used,  which  may  be 
due  largely  to  an  important  property  of  it  to  be  discussed  later  (§  64). 

*  Crelle,  Vol.  VII  (1831),  p.  1.  f  Acta  Mathematica,  Vol.  XIV  (1890),  p.  95. 


146        GEOMETRY  OF  A  SURFACE  ABOUT  A  POINT 

62.  Helicoids.  We  apply  the  preceding  results  in  a  study  of 
an  important  class  of  surfaces  called  the  helicoids.  A  helicoid  is 
generated  by  a  curve,  plane  or  twisted,  which  is  rotated  about  a 
fixed  line  as  axis,  and  at  the  same  time  translated  in  the  direction 
of  the  axis  with  a  velocity  which  is  in  constant  ratio  with  the 
velocity  of  rotation.  A  section  of  the  surface  by  a  plane  through 
the  axis  is  called  a  meridian.  All  the  meridians  are  equal  plane 
curves,  and  the  surface  can  be  generated  by  a  meridian  moving 
with  the  same  velocities  as  the  given  curve.  The  particular  motion 
described  is  called  helicoidal  motion,  and  so  we  may  say  that  any 
helicoid  can  be  generated  by  a  plane  curve  with  helicoidal  motion. 

In  order  to  determine  the  equations  of  a  helicoid  in  parametric 
form,  we  take  the  axis  of  rotation  for  the  2-axis,  and  let  u  denote 
the  distance  of  a  point  of  the  surface  from  the  axis,  and  v  the  angle 
made  by  the  plane  through  the  point  and  the  axis  with  the  #z-plane 
in  the  positive  direction  of  rotation.  If  the  equation  of  the  gen 
erating  curve  in  any  position  of  its  plane  is  z  =  <f>  (w),  the  equations 
of  the  surface  are 

(81)  x  =  u  cos  v,         y  =  u  sin  v,         z  =  </>  (u)  +  av, 

where  a  denotes  the  constant  ratio  of  the  velocities ;  it  is  called 
the  parameter  of  the  helicoidal  motion.  When,  in  particular,  a  is 
zero,  these  equations  define  any  surface  of  revolution.  Moreover, 
when  <£  (u)  is  a  constant,  the  curves  v  =  const,  are  straight  lines 
perpendicular  to  the  axis,  and  so  the  surface  is  a  right  conoid. 
It  is  called  the  right  helicoid, 

By  calculation  we  obtain  from  (81) 

(82)  ^  =  l  +  c/)/2,         F=a<t>',         G=u2+a2, 

where  the  accent  indicates  differentiation  with  respect  to  u.  From 
the  method  of  generation  it  follows  that  the  curves  v  =  const,  are 
meridians,  and  u  =  const,  are  helices  on  the  helicoids,  and  circles 
on  surfaces  of  revolution.  From  (82)  it  is  seen  that  these  curves 
form  an  orthogonal  system  only  on  surfaces  of  revolution  and  on 
the  right  helicoid.  Moreover,  from  (57)  it  is  found  that  the  geo 
desic  curvature  of  the  meridians  is  zero  only  when  a  is  zero  or  <£' 
is  a  constant.  In  the  latter  case  the  meridian  is  a  straight  line 
perpendicular  to  the  axis  or  oblique,  according  as  (/>'  is  zero  or  not. 


HELICOIDS  147 

Hence  the  meridians  of  surfaces  of  revolution  and  of  the  ruled 
helicoids  are  geodesies. 

The  orthogonal  trajectories  of  the  helices  upon  a  helicoid  are 
determined  by  the  equation  (cf.  Ill,  31) 

afidu  +  (u2  +  a2)  dv  =  0. 
Hence,  if  we  put  v1  =  /     >2  ^  2  du  +  v, 

J  It  -J-  Q 

the  curves  vx  =  const,  are  the  orthogonal  trajectories,  and  their 
equations  in  finite  form  are  found  by  a  quadrature.  In  terms  of 
the  parameters  u  and  vl  the  linear  element  is 

(83)  t?s2 


u 


As  an  immediate  consequence  of  this  result  we  have  that  the 
helices  and  their  orthogonal  trajectories  on  any  helicoid  form  an 
isothermal  system. 

From  (83)  and  (§  46)  we  have  the  theorem  of  Bour: 

Every  helicoid  is  applicable  to  some  surface  of  revolution,  and 
helices  on  the  former  correspond  to  the  parallels  on  the  latter. 

We  derive  also  the  following  expressions  : 

„  ,r  „     a  sin  v  —  u<f>'  cos  v,  —  (a  cos  v  +  u$!  sin  v),  u 

(o4)      JL,  y,  z  =  -  -  » 

V^2(l  +  f2)+a2 
and 

(85)  />,/>'.!>"= 


From  (84)  it  follows  that  a  meridian  is  a  normal  section  of  a  sur 
face  of  revolution  at  all  its  points,  and  consequently  is  a  line  of 
curvature  (Ex.  7,  p.  140).  This  is  evident  also  from  the  equation 
of  the  lines  of  curvature  of  a  helicoid,  namely 

(86)    a  [1  +  </>'2  +  u<t>'$']  dii2  +  [(u2  +  a2)  M<£"  -  (1  +  $'*)  u^']  dudv 

-a[u2(t>'-2+u2+a2]dv2=Q. 

Moreover,  the  meridians  are  lines  of  curvature  of  those  helicoids, 
for  which  </>  satisfies  the  condition 


148       GEOMETRY  OF  A  SURFACE  ABOUT  A  POINT 


By  integration  this  gives 

<f)  =  Vtf"  —  U2  —  C  log 


When  the  surface  is  the  right  helicoid  the  expressions  for  D 
and  D"  vanish.  Hence  the  meridians  and  helices  are  the  asymp 
totic  lines.  Moreover,  these  lines  form  an  orthogonal  system,  so 
that  the  surface  is  a  minimal  surface  (§  55).  Since  the  tan- 

gent  planes  to  a  surface  along 

an  asymptotic  line  are  its  oscu 
lating  planes,  if  the  surface  is  a 
ruled  minimal  surface,  the  gener 
ators  are  the  principal  normals  of 
all  the  curved  asymptotic  lines. 
But  a  circular  helix  is  the  only 
Bertrand  curve  whose  principal 
normals  are  the  principal  normals 
of  an  infinity  of  curves  (§  19). 
Hence  we  have  the  theorem  of 
Catalan : 

The  right  helicoid  is  the  only 
real  minimal  ruled  surface. 

In  fig.  16  are  represented  the  asymptotic  lines  and  lines  of 
curvature  of  a  right  helicoid. 

For  any  other  helicoid  the  equation  of  the  asymptotic  lines  is 
(87)  ufi'du2-  Zadudv  +  uty'di?  =  0. 

As  the  coefficients  in  (86)  and  (87)  are  functions  of  u  alone,  we 
have  the  theorem  : 

When  a  helicoid  is  referred  to  its  meridians  and  helices,  the  asymp 
totic  lines  and  the  lines  of  curvature  can  be  found  by  quadratures. 

EXAMPLES 

1.  Show  that  the  spherical  representation  of  the  lines  of  curvature  of  a  surface 
of  revolution  is  isothermal. 

2.  The  osculating  planes  of  a  line  of  curvature  and  of  its  spherical  representa 
tion  at  corresponding  points  are  parallel. 

3.  The  angles  between  the  asymptotic  directions  at  a  point  on  a  surface  and 
between  their  spherical  representation  are  equal  or  supplementary,  according  as 
the  surface  has  positive  or  negative  curvature  at  the  point. 


FIG.  16 


GENERAL  EXAMPLES  149 

4.  Show  that  the  helicoidal  surface 

x  =  u  cos  v,        y  =  u  sin  v,         z  =  bv 
is  minimal. 

5.  The  total  curvature  of  a  helicoid  is  constant  along  a  helix. 

6.  The  orthogonal  trajectories  of  the  helices  upon  a  helicoid  are  geodesies. 

7.  If  the  fundamental  functions  E,  F,  G  of  a  surface  are  functions  of  a  single 
parameter  w,  the  surface  is  applicable  to  a  surface  of  revolution. 

8.  Find  the  equations  of  the  helicoid  generated  by  a  circle  of  constant  radius 
whose  plane  passes  through  the  axis  and  the  lines  of  curvature  on  the  surface  ;  also 
find  the  equations  of  the  surface  in  terms  of  parameters  referring  to  the  meridians 
and  their  orthogonal  trajectories. 

GENERAL  EXAMPLES 

1.  If  a  pencil  of  planes  be  drawn  through  a  tangent  MT  to  a  surface,  and  if 
lengths  be  laid  off  on  the  normals  at  M  to  the  sections  of  the  surface  by  these 
planes  equal  to  the  curvature  of  the  sections,  the  locus  of  the  end  points  is  a 
straight  line  normal  to   the  plane   determined  by  MT  and  the   normal  to  the 
surface  at  M. 

2.  If  P  is  a  point  of  a  developable  surface,  P0  the  point  where  the  generator 
through  P  touches  the  edge  of  regression,  t  the  length  PoP,  p  and  r  the  radii  of 
curvature  and  torsion  of  the  edge  of  regression,  then  the  principal  radii  of  the 
surface  are  given  by  -i  -i 

—  =  0,        -  =  -  • 

3.  For  the  surface  of  revolution  of  a  parabola  about  its  directrix,  the  principal 
radii  are  in  constant  ratio. 

4.  The  equations  x  =  a  cos  it,   y  =  asinw,   z  =  uv  define  a  family  of  circular 
helices  which  pass  through  the  point  A  (a,  0,  0)  of  the  cylinder ;  each  helix  has  an 
involute  whose  points  are  at  the  distance  c  from  A  (cf.  I,  106).    Find  the  surface 
which  is  the  locus  of  these  involutes ;  show  that  the  tangents  to  the  helices  are 
normal  to  this  surface ;  find  also  the  lines  of  curvature  upon  the  latter. 

5.  The  surfaces  defined  by  the  equations  (cf.  §  25) 

l+p2  +  ?2  =  q*f(y),        x  +  pz=<t>(p) 

have  a  system  of  lines  of  curvature  in  planes  parallel  to  the  xz-plane  and  to  the 
y-axis  respectively. 

6.  The  equations 

y  -  ax  =  0,        a:2  +  y2  +  z2  -  2  px  -  a2  =  0, 

where  a  and  ft  are  parameters,  define  all  the  circles  through  the  points  (0,  0,  a), 
(0,0,  —a).  Show  that  the  circles  determined  by  a  relation  ft=f(a)  are  the 
characteristics  of  a  family  of  spheres,  except  when  f(a)  is  a  linear  function ; 
also  that  the  circles  are  lines  of  curvature  on  the  envelope  of  these  spheres. 

7.  If  one  of  the  lines  of  curvature  of  a  developable  surface  lies  upon  a  sphere, 
the  other  nonrectilinear  lines  of  curvature  lie  on  concentric  spheres. 


150        GEOMETRY  OF  A  SURFACE  ABOUT  A  POINT 

8.  If  P  is  a  point  on  a  surface,  P0  the  center  of  normal  curvature  of  the  line 
bisecting  the  angle  between  the  lines  of  curvature,  and  PI,  P2  the  centers  of  normal 
curvature  in  two  directions  equally  inclined  to  the  first,  then  the  four  points 
P,  PI,  PO,  PZ  form  a  harmonic  range. 

9.  If  EI,  Ra,  R-s,  •  •  • ,  Rm  denote  the  radii  of  normal  curvature  of  m  sections  of 
a  surface  which  make  equal  angles  2  tr/m  with  one  another,  and  m  >  2,  then 

I /-I  V--1  +  J_\  =  * /i  +  IV 

m  \Ri      R2  RJ      2  Vx  T  PJ 

10.  If  the  Dupin  indicatrix  at  a  point  P  of  a  surface  is  an  ellipse,  and  through 
either  one  of  the  asymptotes  of  its  focal  hyperbola  two  planes  be  drawn  perpen 
dicular  to  one  another,  their  intersections  with  the  tangent  plane  are  conjugate 
directions  on  the  surface. 

1 1 .  All  curves  tangent  to  an  asymptotic  line  at  a  point  M ,  and  whose  osculating 
planes  are  not  tangent  to  the  surface  at  3f,  have  M  for  a  point  of  inflection. 

12.  The  normal  curvature  of  an  orthogonal  trajectory  of  an  asymptotic  line  is 
equal  to  the  mean  curvature  of  the  surface  at  the  point  of  intersection. 

13.  The  surface  of  revolution  whose  equations  are 

x  =  u  cos  w,        y  =  u  sin  w,        z  =  a  log  (u  4-  vV2  —  a2) 

is  generated  by  the  rotation  of  a  catenary  about  its  axis ;  it  is  called  the  catenoid. 
Show  that  it  is  the  only  minimal  surface  of  revolution. 

14.  When  the  osculating  plane  of  a  line  of  curvature  makes  a  constant  angle 
with  the  tangent  plane  to  the  surface,  the  line  of  curvature  is  plane. 

15.  A  plane  line  of  curvature  is  represented  on  the  unit  sphere  by  a  circle. 

16.  The  cylinder  whose  right  section  is  the  curve  defined  by  the  intrinsic  equa 
tion  p  =  a  -  s2/6,  where  a  and  b  are  positive  constants,  has  the  characteristic  prop 
erty  that  upon  it  lie  curves  of  curvature  ^la  +  b ,  whose  geodesic  curvature  is 
I/Void.  ^  a26 

17.  When  a  surface  is  referred  to  an  orthogonal  system  of  lines,  and  the  radii  of 
geodesic  curvature  of  the  curves  v  =  const,  and  u  -  const.  are  p^,  pgv  respectively, 
the  geodesic  curvature  of  the  curve  which  makes  an  angle  0Q  with  the  lines  v  =  const, 
is  given  by  1  _  dd0      cos00      sin  00  > 

Pg  ~~   dS  Pgu  Pgv 

18.  When  a  surface  is  referred  to  an  orthogonal  system  of  lines,  and  pvi  s 
denote  the  radius  of  geodesic  curvature  and  the  arc  for  one  system  of  isogonal 
trajectories  of   the   parametric  lines,    and  pj,  8'  the  similar  functions  for  the 
orthogonal  trajectories  of  the  former,  then  whatever  be  the  direction  of  the  first 
curves  the  quantity  —  —  4.  —  —  to  constant  at  a  point. 

19.  If  p  and  p'  denote'the  radii  of  first  curvature  of  a  line  of  curvature  and  its 
spherical  representation,  and  also  p,  and  p'g  the  radii  of  geodesic  curvature  of  these 
curves,  then  fis      dff          fa      dtr 

7  =  7'        P~8~P?' 
where  ds  and  d<r  are  the  linear  elements  of  the  curves. 


GENERAL  EXAMPLES  151 

20.  When  a  surface  is  referred  to  its  lines  of  curvature,  and  00,  #o  denote  the 
angles  which  a  curve  on  the  surface  and  its  spherical  representation  make  with 
the  curves  v  =  const.,  the  radii  of  geodesic  curvature  of  these  curves,  denoted  by 
pg  and  pg  respectively,  are  in  the  relation 

ds  dcr 

ddo  —  —  =  d&o  --  -  • 
Py  Pg 

21.  When  the  curve 

x=f(u)cosu,        y=f(u)sinu,        z  =  - 

is  subjected  to  a  helicoidal  motion  of  parameter  a  about  the  z-axis,  the  various 
positions  of  this  curve  are  orthogonal  trajectories  of  the  helices,  and  also  geodesies 
on  the  surface. 

22.  When  a  curve  is  subjected  to  a  continuous  rotation  about  an  axis,  and  at 
the  same  time  to  a  homothetic  transformation  with  respect  to  a  point  of  the  axis, 
such  that  the  tangent  to  the  locus  described  by  a  point  of  the  curve  makes  a  con 
stant  angle  with  the  axis,  the  locus  of  the  resulting  curves  is  called  a  spiral  surface. 
Show  that  if  the  z-axis  be  taken  for  the  axis  of  rotation  and  the  origin  for  the  center 
of  the  transformation,  the  equations  of  the  surface  are  of  the  form 

x  =  f(u)  ehv  cos  (u  +  v)  ,        y  =  f(u)  ehv  sin  (u  +  v)  ,        z  =  0  (u)  e*v, 
where  h  is  a  constant. 

23.  A  spiral  surface  can  be  generated  in  the  following  manner:   Let  C  be  a 
curve,  I  any  line,  and  P  a  point  on  the  latter  ;  if  each  point  M  on  C  describes  an 
isogonal  trajectory  of  the  generators  on  the  circular  cone  with  vertex  P  and  axis  I 
in  such  a  way  that  the  perpendicular  upon  I,  from  the  moving  point  M  ,  revolves  about  I 
with  constant  velocity,  the  locus  of  these  curves  is  a  spiral  surface  (cf  .  Ex.  5,  §  33). 

24.  Show  that  the  orthogonal  trajectories  of  the  curves  u  =  const.,  in  Ex.  22, 
can  be  found  by  quadratures,  and  that  the  linear  element  can  be  put  in  the  form 


where  A  is  a  function  of  a  alone. 

25.  Show  that  the  lines  of  curvature,  minimal  lines,  and  asymptotic  lines  upon 
a  spiral  surface  can  be  found  by  quadrature. 


CHAPTER  V 


FUNDAMENTAL  EQUATIONS.    THE  MOVING  TRIHEDRAL 

63.  Christoffel  symbols.  In  this  chapter  we  derive  the  necessary 
and  sufficient  equations  of  condition  to  be  satisfied  by  six  func 
tions,  E,  F,  G  ;  D,  D\  D",  in  order  that  they  may  be  the  fundamental 
quantities  for  a  surface. 

For  the  sake  of  brevity  we  make  use  of  two  sets  of  symbols, 
suggested  by  Christoffel,*  which  represent  certain  functions  of  the 
coefficients  of  a  quadratic  differential  form  and  their  derivatives  of 
the  first  order.  If  the  differential  form  is 

andu2 -f-  2  a^du^du^  +  c 
the  first  set  of  symbols  is  defined  by 


R&l    !/^  +  ^_<HA 

[l  J      2\duk      du(       duj 


where  each  of  the  subscripts  i,  k,  I  has  one  of  the  values  1  and  2.f 
From  this  definition  it  follows  that 


When  these  symbols  are  used  in  connection  ,with  the  first  fun 
damental  quadratic  form  of  a  surface  ds2  =  E du2  +  2F dudv  +  G  dv2, 
they  are  found  to  have  the  following  significance : 


(1) 


[iriia*  [""L^ia* 

L  1  J     2Su  L  2  J     Su      2  0» 


2dv 


I    ~l     I         Q/»*          O       ,*  I    O    I         0 

L  J-   I       cv       &  cu         L  ^  J      ^ 


2  J     2  di< 

2~dv 


*  Crelle,  Vol.  LXX,  pp.  241-245. 

t  This  equation  defines  these  symbols  for  a  quadratic  form  of  any  number  of  vari 
ables  wi,  •  •  • ,  un.   In  this  case  i,  k,  I  take  the  values  1,  •  •  • ,  n. 

152 


CHRISTOFFEL  SYMBOLS  153 

The  second  set  of  symbols  is  defined  by  the  equation 


where  Avl  denotes  the  algebraic  complement  of  avl  in  the  discrimi 
nant  ana22  —  a22  divided  by  the  discriminant  itself.  With  reference 
to  the  first  fundamental  quadratic  form  these  symbols  mean 

4i--Jv     ^=:^'     A»=-!r*' 

and 

du          dv  du       fill  du         dv  du 


f)     TT'2  '  I         n         I  ,"»      Ti-O 


ri2\     ^g^       du  I12\_ 

llJ  2^        '  l2/~ 


du          dv 


—f"—  —  G—±  2  G  — 

^v          3t*  dv    f221          dv   '      du  '          dv 


From  these  equations  we  derive  the  following  identities  : 


With  the  aid  of  these  identities  we  derive  from  (III,  15,  16)  the 
expressions 

«•>£—  (MVKff}>  £—  Sm-K?))- 

f^l 

From  the  above  definition  of  the  symbols  <!      >  WQ  obtain  the 
following  important  relation  : 


64.    The  equations  of  Gauss  and  of  Codazzi.    The  first  two  of 
equations  (IV,  10)  and  the  equation 


154 


FUNDAMENTAL  EQUATIONS 


form  a  consistent  set  of  equations  linear  in  — ,  -£»  — ,  and  the 

2      du2    du2    du2 

determinant  is  equal  to  H.    Solving  for  — -,  we  get 


similar  equations  hold  for  y  and  z.    Proceeding  in  like  manner  with 
the  other  equations  (IV,  10)  and 


(6) 


we  get  the  following  equations  of  G-auss  : 


a2*  _ri2\az    fiaiat    ^ 
awa«    li/a*    la/a* 


For  convenience  of  reference  we  recall  from  §  48  the  equations 


(8) 


dX_FD'—GDdx      FD—ED'dx 

~du=      H2      a^4      n2      dv' 

dX  __  FD"-GD'  dx      FD'-ED"  dx 
3v  ~          H2         du  H2         dv 


The  conditions  of  integrability  of  the  Gauss  'equations  (7)  are 


du\dudv 


— 

dv  \dudv       du\dv2 


By  means  of  (7)  and  (8)  these  equations  are  reducible  to  the  forms 
<*x       v—  o  ^  4.  A  ?E  _i_    x—  0 

1 dv        1  2  du       2  v 


(9) 


a 


where  av  a2,  •  •  >,  c2  are  determinate  functions  of  E,  F,  G  ;  D,  D',  D" 
and  their  derivatives.    Since  equations  similar  to  (9)  hold  for  y  and 
z,  we  must  have 
(10)         «!=0,     a2=0,     &j=0,     62=0,     ^=0,     <?2=0. 


EQUATIONS  OF  GAUSS  AND  OF  CODAZZI  155 

When  the  expressions  for  ar  a2,  b^  and  52  are  calculated,  it  is 
found  that  the  first  four  equations  are  equivalent  to  the  following  : 

12 


12\     jMlll      |12jfl2j      fill  J221 

d  /in     d  ri2i  ,  rni/121  ,  rnif22 


fi2i  fill     ri2i2 
HiAaJ-ta-J' 

221      d   M21  ,  /221 /121  ,  /221  Ml 


f!2\      a  /22\  ,   fl2\fl2\     f22\/ll 


When  the  expressions  for  the  Christoffel  symbols  are  substituted 
in  these  equations  the  latter  reduce  to  the  single  equation 


Z?D"-jP'a=    1     f  g  r  F    dE__  1  3G1 
H2         ~2N  \du  \_EH  dv      H  du\ 


j^r  2  dF    i      _ 


dv  IH  du      H  dv       EH  du_ 

This  equation  was  discovered  by  Gauss,  and  is  called  the  G-auss 
equation  of  condition  upon  the  fundamental  functions.  The  left- 
hand  member  of  the  equation  is  the  expression  for  the  total  curva 
ture  of  the  surface.  Hence  we  have  the  celebrated  theorem  of 
Gauss  * : 

The  expression  for  the  total  curvature  of  a  surface  is  a  function 
of  the  fundamental  coefficients  of  the  first  order  and  of  their  deriva 
tives  of  the  first  and  second  orders. 

When  the  expressions  for  c^  and  c2  are  calculated,  we  find  that 
the  last  two  of  equations  (10)  are 


(13) 


v 


du     "  dv 


*L.c.,  p.  20. 


156  FUNDAMENTAL  EQUATIONS 

These  are  the  Codazzi  equations,  so  called  because  they  are  equiva 
lent  to  the  equations  found  by  Codazzi  *  ;  however,  it  should  be 
mentioned  that  Mainardi  was  brought  to  similar  results  some 
what  earlier.  f  It  is  sometimes  convenient  to  have  these  equations 
written  in  the  form 


(13') 


D       d  D'  ,  f22\D      „  f!2\D'      fin 
~  " 


~~ 


which  reduce  readily  to  (13)  by  means  of  (3). 

With  the  aid  of  equations  (7)  we  find  that  the  conditions  of 
integrability  of  equations  (8)  and  similar  ones  in  Y  and  Z  reduce 
to  (13). 

From  the  preceding  theorem  and  the  definition  of  applicable 
surfaces  (§  43)  follows  the  theorem  : 

Two  applicable  surfaces  have  the  same  total  curvature  at  corre 
sponding  points. 

As  a  consequence  we  have  : 

Every  surface  applicable  to  a  plane  is  the  tangent  surface  of  a 
twisted  curve. 

For,  when  a  surface  is  applicable  to  a  plane  'its  linear  element  is 
reducible  to  ds'2=  du'2+  dv\  and  consequently  its  total  curvature 
is  zero  at  every  point  by  (12).  From  (IV,  73)  it  follows  that 


'2 


Hence  X,  Y,  Z  are  functions  of  a  single  parameter,  and  therefore 
the  surface  is  the  tangent  surface  of  a  twisted  curve  (cf.  §  27). 
Incidentally  we  have  proved  the  theorem : 

When  K  is  zero  at  all  points  of  a  surface  the  latter  is  developable, 
and  conversely. 

*  Sulle  coordinate  curvilinee  d'una  superficie  e  dello  spazio,  Annali,  Ser.  3,  Vol.  II 
(18(W) ,  p.  269. 

t  Giornale  dell' Istituto  Lombardo,  Vol.  IX,  p.  395. 


FUNDAMENTAL  THEOREM 


157 


65.  Fundamental  theorem.    When   the  lines  of  curvature   are 
parametric,  the  Gauss  and  Codazzi  equations  (12),  (13)  reduce  to 


(14) 


DP" 


_ 


G      dv 


2_  (D"\  _  ^ 
~ 


E    du 


The  direction-cosines  of  the  tangents  to  the  parametric  curves, 
v  =  const,  and  u  =  const.,  have  the  respective  values 


(15) 


2» 


By  means  of  equations  (7)  and  (8)  we  find 


(16)    . 


du 


(  du 


D" 

V£   ' 


ax 


and  similar  equations  obtained  by  replacing  X^  X^  X  by  Yv  T2,  Y 
respectively,  and  by  Z^  Z^,  Z.    From  (15)  we  have 


(17) 


We  proceed  to  the  proof  of  the  converse  theorem : 
Given  four  functions,  E,  G,  D,  D",  satisfying  equations  (14);  there 
exists  a  surface  for  which  E,  0,  G ;   Z>,  0,  D"  are  the  fundamental 
quantities  of  the  first  and  second  order  respectively. 


=  C^EX 
=  CVflY 
=  CVEZ 


158  FUNDAMENTAL  EQUATIONS 

In  the  first  place  we  remark  that  all  the  conditions  of  integra- 
bility  of  the  equations  (16)  are  satisfied  in  consequence  of  (14). 
Hence  these  equations  admit  sets  of  particular  solutions  whose 
values  for  the  initial  values  of  u  and  v  are  arbitrary.  From  the 
form  of  equations  (16)  it  follows  (cf.  §  13)  that,  if  two  such  sets 
of  particular  solutions  be  denoted  by  X^  Xz,  X  and  Y^  F2,  F,  then 

I  XI  +  XI  +  X*  =  const., 
Y?+  F2  +  F2=  const., 
X1 Fx  +  X2  Y2  +  XY=  const. 

From  the  theory  of  differential  equations  we  know  that  there  exist 
three  particular  sets  of  solutions  X^  Xz,  X\  Fp  F2,  F;  Z^  Z^  Z, 
which  for  the  initial  values  of  u  and  v  have  the  values  1,  0,  0;  0, 1,  0; 
0,  0, 1.  In  this  case  equations  (18)  become 

•X12  +  X22  +  ^2  =  1, 

(19)  F2+F22+ F2  =  l, 
XiYl+X2Y2  +  XY=Q, 

which  are  true  for  all  values  of  u  and  v.  In  like  manner  we  have 
(19') 

From  (16)  it  follows  that  the  expressions  in  the  right-hand  mem 
bers  of  (17)  are  exact  differentials,  and  that  the  surface  denned  by 
these  equations  has,  for  its  linear  element  and  its  second  quadratic 
form,  the  expressions 

(20)  Edu?+G  dv\         D  dy?  +  D"dv2 
respectively. 

Suppose,  now,  that  we  had  a  second  system  of  three  sets  of 
solutions  of  equations  (16)  satisfying  the  conditions  (19),  (19'). 
By  a  motion  in  space  we  could  make  these  X's,  F's,  and  Z's  equal 
to  the  corresponding  ones  of  the  first  system  for  the  initial  values 
of  u  and  v.  But  then,  because  of  the  relations  similar  to  (18),  they 
would  be  equal  for  all  values  of  u  and  v,  as  shown  in  §  13.  Hence, 
to  within  a  motion  in  space,  a  surface  is  determined  by  two  quad 
ratic  forms  (20).  As  in  §  13,  it  can  be  shown  that  the  solution  of 
equations  (16)  reduces  to  the  integration  of  an  equation  of  Riccati. 


FUNDAMENTAL  THEOEEM  159 

Later  *  we  shall  find  that  the  direction-cosines  of  any  two  per 
pendicular  lines  in  the  tangent  plane  to  a  surface,  and  of  the  normal 
to  the  surface,  satisfy  a  system  of  equations  similar  in  form  to  (16). 
Moreover,  these  equations  possess  the  property  that  sets  of  solu 
tions  satisfy  the  conditions  (18)  when  the  parametric  lines  are  any 
whatever.  Hence  the  choice  of  lines  of  curvature  as  parametric 
lines  simplifies  the  preceding  equations,  but  the  result  is  a  general 
one.  Consequently  we  have  the  following  fundamental  theorem: 

When  the  coefficients  of  two  quadratic  forms, 

Edu*  +  2  Fdudv  +  G  dv\          Ddu2+2  D'dudv  +  D"dv\ 

satisfy  the  equations  of  Gauss  and  Codazzi,  there  exists  a  surface, 
unique  to  within  its  position  in  space,  for  which  these  forms  are 
respectively  the  first  and  second  fundamental  quadratic  forms  ;  and 
the  determination  of  the  surface  requires  the  integration  of  a  Riccati 
equation  and  quadratures. 

From  (III,  3),  (5)  and  (6),  it  follows  that  if  E,  F,  G;  D,  D',  D"  are  the  funda 
mental  functions  for  a  surface  of  coordinates  (x,  y,  z),  the  surface  symmetric  with 
respect  to  the  origin,  that  is,  the  surface  with  the  coordinates  (—  x,  —  y,  —  z),  has 
the  fundamental  functions  E,  F,  G;  -  D,  -  D',  -  D".  Moreover,  in  consequence 
of  the  above  theorem,  two  surfaces  whose  fundamental  quantities  bear  such  a  rela 
tion  can  be  moved  in  space  so  that  they  will  be  symmetric  with  respect  to  a  point. 
Two  surfaces  of  this  kind  will  be  treated  as  the  same  surface. 

EXAMPLES 

1.  When  the  lines  of  curvature  of  a  surface  form  an  isothermal  system,  the 
surface  is  said  to  be  isothermic.    Show  that  surfaces  of  revolution  are  isothermic. 

2.  Show  that  the  hyperbolic  paraboloid 

a  b  uv 

x  =  -(t*  +  t>),       y  =  -(*-»),       *  =  - 

is  isothermic. 

3.  When  a  surface  is  isothermic,  and  the  linear  element,  expressed  in  terms  of 
parameters  referring  to  the  lines  of  curvature,  is  ds2  =  \2(du2  +  dv2),  the  equations 
of  Codazzi  and  Gauss  are  reducible  to 

Pl     i     dp2        a        _  PZ     i     api 


4.  Find  the  form  of  equations  (11),  (13)  when  the  surface  is  defined  in  terms  of 
symmetric  coordinates  (cf  .  §  39). 

*  Cf  .  §  69.  Consult  also  Scheffers,  Vol.  II,  pp.  310  et  seq.  ;  Bianchi,  Vol.  I,  pp.  122-124. 


160  FUNDAMENTAL  EQUATIONS 

5.  Show  that  K  is  equal  to  zero  for  the  tangent  surface  of  a  twisted  curve, 
taking  the  linear  element  of  the  latter  in  the  form  (105),  §  20. 

6.  Show  that  the  total  curvature  of  the  surface  of  revolution  of  the  tractrix 
about  its  axis  is  negative  and  constant. 

7.  Establish  the  following  formulas,  in  which  the  differential  parameters  are 
formed  with  respect  to  the  form  Edu?  +  2Fdudv  +  Gdv2: 


)=~- 

where  the  quantities  have  the  same  significance  as  in  §  65. 


8.  Deduce  the  identity  A2x 


=  (  —  1  —  )  JT, 


and  show  therefrom  that  the  curves  in  which  a  minimal  surface  is  cut  by  a 
family  of  parallel  planes  and  the  orthogonal  trajectories  of  these  curves  form 
an  isothermal  system. 

66.  Fundamental  equations  in  another  form.  We  have  seen  in 
§  61  that  if  X,  F,  Z  denote  the  direction-cosines  of  the  normal  to 
a  surface,  the  direction-cosines  of  the  normal  to  the  spherical  rep 
resentation  of  the  surface  are  eX,  eF,  eZ,  where  e  is  ±  1  according 
as  the  curvature  of  the  surface  is  positive  or  negative.  If,  then, 
the  second  fundamental  quantities  for  the  sphere  be  denoted  by 
A  ^',  3",  we  have 

(21)  £=-€<£          ^'  =  -e^,          ,&"  =  -e^ 

so  that  for  the  sphere  equations  (7)  become 


rnvax_ 

12J  dv 


2 

2X      f 

J 


2X     f  12V 

+  l2J 


(22) 

F-^x; 

where  the  Christoffel  symbols  \T?\  are  formed  with  respect  to 
the  linear  element  of  the  spherical  representation,  namely 

The  conditions  of  integrability  of  equations  (22)  are  reducible 
by  means  of  the  latter  to 

—  =  0, 


EQUATIONS  OF  CODAZZI  161 

where  A.,  A2J  B^  B2  are  the  functions  obtained  from  the  quantities 

DD"—D'2 
av  a2,  blt  b2  respectively  of  §  64  by  replacing  — >  E,  F,  G 

by  1,  €,  &,  ^respectively.  Since  the  above  equations  must  be  sat 
isfied  by  Y  and  Z,  the  quantities  A^  A2,  Bv  Bz  must  be  zero.  This 
gives  the  single  equation  of  condition 

(24}  J_rA/^^__L^V-^-  — --  — --^-— ^1  =  1 
J  '   2  ft  [du  \€ft  dv      ft  du  )     to  W  fa      ft  2v      &ft  du)\ 

Moreover,  the  Codazzi  equations  (13')  become,  in  consequence 
of  (21), 


(26). 

which  vanish  identically. 


o       \     //"/          o      V    //•  /    '      I     1  //"          ~   *     ~*      *        ' f     '      '     ~*       *       "" 

3w  W  /       £v  \///        \.\jft 


dx          dx 
If  equations  (IV,  13)  be  solved  for  —  and  — »  we  get 

du          cv 


f  dx  _ 

" 


(26) 


. i_ 

dv  ft*         du 


By  means  of  equations  (22)  the  condition  of  integrability  of  these 
equations,  namely  g  /^\       ^ 


and  similar  conditions  in  y  and  2,  reduce  to 

(27)  - 

OU  v^ 

^Hu      dv 

Hence  two  quadratic  forms 

(odu2  -f  2  &  dudv  +  £  dv2,         D  du2  +2Z>'  dudv  +  D"dv2, 

whose  coefficients  satisfy  the  conditions  (24),  (27),  may  be  taken 
as  the  linear  element  of  the  spherical  representation  of  a  surface 
and  as  the  Si3cond  quadratic  form  of  the  latter.  When  X,  F,  Z  are 


162 


FUNDAMENTAL  EQUATIONS 


known,  the  cartesian  coordinates  of  the  surface  can  be  found  by 
quadratures  (26) ;  however,  the  determination  of  the  former  requires 
the  solution  of  a  Riccati  equation. 


If  the  equations 

D  =  _^fc*xt 


be  differentiated  with  respect  to  u  and  v,  the  resulting  equations  may  be  reduced 
by  means  of  (7)  and  (22)  to  the  form :  * 

55    (ii 

cu       (  1 

—  =  [I2l 
a»     c  i 


cD' 

cu 


^-=    Jl*   JX  + 


cv 


cu 


cv 


(V 


D 


D" 


2)  -f 
D  + 
D  -f 


11 
2 

12 
2 
12 
2 

22 )' 
2 


D". 


67.  Tangential  coordinates.  Mean  evolute.  A  surface  may  be 
looked  upon  not  only  as  the  locus  of  a  point  whose  position 
depends  upon  two  parameters,  but  also  as  the  envelope  of  its 
tangent  planes.  This  family  of  planes  depend*  upon  one  or  two 
parameters  according  as  the  surface  is  developable  or  not.  We 
considered  the  former  case  in  §  27,  and  now  take  up  the  latter. 

If  W  denotes  the  algebraic  distance  from  the  origin  to  the  tan 
gent  plane  to  a  surface  S  at  the  point  M(x,  y,  z),  then 
(29)  W=xX+yY+zZ. 

If  this  equation  is  differentiated  with   respect  to   u   and  v,  the 
resulting  equations  are  reducible,  in  consequence  of  (IV,  3),  to 

X  dW 


*Cf.  Bianchi,  Vol.  I,  p.  157. 


TANGENTIAL  COORDINATES 


163 


The  three  equations  (29),  (30)  are  linear  in  #,  y,  z,  and  in  con 
sequence  of  (IV,  79,  80)  their  determinant  is  equal  to  e/£  Hence 
we  have 


W       Y       Z 

e 

dW     cY     dZ 

x  =  — 

tt 

cu      du      du 

dW     cY     dZ 

dv       dv      dv 

and  similar  expressions  for  y  and  z.    From  (IV,  11)  we  deduce  the 

identities                 dz         dY      e  /      ^dX  ,    ~dX\ 
Y Z —  =  TH  — ex1 r  <o )» 

01  du          du       rf  \         du  dv  / 

(dl)  a-         y.v          ,         ~Y         zr< 

-rrC£  rrV^  €     I  C^\-  j^G-A- 

By  means    of    these    equations   the    above    expression    for   x   is 
reducible  to 


cv 
Hence  we  have 

(32)  x  =  WX+k[(W,X),    y  =  WY+k((W,Y),     z  =  WZ+  &((W,Z], 

the  differential  parameters  being  formed  with  respect  to  (23). 

Conversely,  if  we  have  four  functions  X,  F,  Z,  W  of  u  and  i>, 
such  that  the  first  three  satisfy  the  identity 

(33)  x'2+r2+^2  =  l, 

equations  (32)  define  the  surface  for  which  X,  F,  Z  are  the  direction- 
cosines  of  the  tangent  plane,  and  W  is  the  distance  of  the  latter  from 
the  origin.  For,  from  (33),  we  have 


dv 


=  0, 


in  consequence  of  which  and  formulas  (22)  we  find  from  (32)  that 
>TA     dx 


Moreover,  equation  (29)  also  follows  from  (32).  Hence  a  surface 
is  completely  defined  by  the  functions  X,  F,  Z,  W,  which  are 
called  the  tangential  coordinates  of  the  surface.* 


*  Cf .  Weingarten,  Festschrift  der  Technischen  Hochschule  zu  Berlin  (1884) ;  Bianchi, 
Vol.  I,  pp.  172-174 ;  Darboux,  Vol.  I,  pp.  234-248. 


164 


FUNDAMENTAL  EQUATIONS 


When  equations  (30)  are  differentiated,  we  obtain 

ffw 


_ 

dv* 
By  means  of  (22),  (29),  and  (30)  these  equations  are  reducible  to 


(34) 


D'  =  - 


du2 

tfW 


\_dudv 


When  these  expressions  for  D,  A'  &"  are  substituted  in  the 
expression  (IV,  77)  for  p^+  p2,  the  latter  becomes 


By  means  of  (25)  this  equation  can  be  written  in  the  form 

(35)  />14-^2  =  -(A;TF4-2^), 

where  the  differential  parameter  is  formed  with  respect  to  the 

linear  element  (23)  of  the  sphere. 

Moreover,  if  A^20  denotes  the  following  expression, 


_  r22y^__ 
1  1  J  an 


2  J 
12 


12 


it  follows  from  (34)  that 

(3T) 


MEAN  EVOLUTE  165 

In  passing  we  shall  prove  that  A220  is  a  differential  parameter 
by  showing  that  it  is  expressible  in  the  form 


(38) 


Without  loss  of  generality  we  take 
(39)  Edu*+Gdv2 

as  the  quadratic  form,  with  respect  to  which  these  differential 
parameters  are  formed.    Then 

1  1    /I  dG      1  dE\  1   dE 


dv  \du 

u  =  -  F£ 

By  substitution  we  find 
_ 


Since  the  terms  in  the  right-hand  member  are  differential  param 
eters,  their  values  are  independent  of  the  choice  of  parameters 
u  and  v,  in  terms  of  which  (39)  is  expressed.  Hence  equation  (38) 
is  an  identity. 

The  coordinates  #0,  y0,  ZQ  of  the  point  on  the  normal  to  a  sur 
face  halfway  between  the  centers  of  principal  curvature  have 
the  expressions 


The  surface  £0  enveloped  by  the  plane  through  this  point,  which 
is  parallel  to  the  tangent  plane  to  the  given  surface,  is  called  the 
mean  evolute  of  the  latter. 

If  W0  denotes  the  distance  from  the  origin  to  this  plane,  we  have 

(40)  W,=  ZXf9=W+^(p1+Pt). 

By  means  of  (35)  this^may  be  written 

(41)  TFO=-JA;TF. 


166  FUNDAMENTAL  EQUATIONS 

EXAMPLES 

1.  Derive  the  equations  of  the  lines  of  curvature  and  the  expressions  for  the 
principal  radii  in  terms  of  W,  when  the  parametric  lines  on  the  sphere  are 

(i)  meridians  and  parallels  ; 
(ii)  the  imaginary  generators. 

Show  that  in  the  latter  case  the  curves  corresponding  to  the  generators  lie  sym 
metrically  with  respect  to  the  lines  of  curvature. 

2.  Let  Wi  and  W2  denote  the  distances  from  the  origin  to  the  planes  through 
the  normal  to  a  surface  and  the  tangents  to  the  lines  of  curvature  v  =  const.  , 
u  =  const,  respectively,  so  that  we  have 

Wi  =  xXl  +  yYi  +  zZi,  W2  =  xX2  +  yYz  +  zZ2. 

Show  that 


Pi 


the  differential  parameters  being  formed  with  respect  to  Edu*  +  2  Fdudv  + 
3.  If  2  q  =  x2  +  yz  +  z2,  then  we  have 


4.  Show  that  when  the  lines  of  curvature  are  parametric 


=  = 

Pi  cu       cu  P2  £v  ~    dv 


5.  The  determination  of  surfaces  whose  mean  evolute  is  a  point  is  the  same 
problem  as  finding  isothermal  systems  of  lines  on  the  sphere. 

68.  The  moving  trihedral.  The  fundamental,  equations  of  con 
dition  may  be  given  another  form,  in  which  they  are  frequently 
used  by  French  writers.  In  deriving  them  we  refer  the  surface  to 
a  moving  set  of  rectangular  axes  called  the  trihedral  T.  Its  ver 
tex  M  is  a  point  of  the  surface,  the  a^-plane  is  tangent  to  the 
surface  at  M,  and  the  positive  2-axis  coincides  with  the  positive 
direction  of  the  normal  to  the  surface  at  M.  The  position  of  the 
x-  and  ?/-axes  is  determined  by  the  angle  U  which  the  tangent  to 
the  curve  v  =  const,  through  M  makes  with  the  z-axis,  U  being  a 
given  function  of  u  and  v. 

In  Chapter  I  we  considered  another  moving  trihedral,  consisting 
of  the  tangent,  principal  normal,  and  binormal  of  a  twisted  curve. 
Let  us  associate  such  a  trihedral  with  the  curve  v  —  const,  through 


THE  MOVING  TRIHEDRAL 


16T 


M  and  call  it  the  trihedral  tu.  We  have  found  (§  16)  that  the  varia 
tions  of  the  direction-cosines  a',  6',  c'  of  a  line  L,  fixed  in  space, 
with  reference  to  £M,  as  its  vertex  moves  along  the  curve  which 
we  call  (7M,  are  given  by 

(42)  <^=b-,    f  W-  + 
dsu      pu     dsu          \Pu 

where  pu,  ru  denote  the  radii  of  first  and  second  curvature  of  Cu1 
and  dsu  its  linear  element ;  evidently  the  latter  may  be  replaced 
by  V^1 du. 

The  direction-cosines  of  L  with  respect  to  the  trihedral  T  have 
the  values      r  a  _  a>  cos  jj  _  ^  sjn  ^  _  c>  cos  ^  sjn  ^r 

(43)  I  b  —  a'  sin  U  +  (br  sin  wu—  c'  cos  o>M)  cos  ?7, 
[<7  =  6f  cos  o>w-f  c'  sin  o)tt, 

where  wu  is  the  angle  which  the  positive  direction  of  the  z-axis 
makes  with  the  positive  direction  of  the  principal  normal  to  Cu  at  Jf, 
the  angle  being  measured  toward  the  positive  direction  of  the  binor- 
mal  of  Cu.  From  equations  (42)  and  (43)  we  obtain  the  following : 


(44) 


da      ,  db  do  , 

-  =  br  —  cq,     —  =  cp  —  ar,     —  =  aq  —  op, 
du  du  cu 


where  j9,  q,  r  have  the  following  significance  : 


(45) 


p  = 


0  = 


cosU 


1          .       coso) 

)-f  sin  U 


rr/c?&>        1\  rrcoso). 

sm£7(  — 2 I—  cosU 

ds,. 


V 


If,  in  like  manner,  we  consider  the  trihedral  tv  of  the  curve 
u  =  const,  through  M,  denoted  by  Cv,  we  obtain  the  equations 

da      ,  db  do  , 


where  pv  qv  rl  can  be  obtained  from  (45)  by  replacing  Vjg;  Z7,  *M, 
^«^  Pui  Tu  ^7  ^^^  ^»  8v->  ®i;»  Pv»  TV-  AS  ^  denotes  the  angle  which  the 
tangent  to  the  curve  Cv  at  M  makes  with  the  a>axis,  we  have 

(46)  V-U=G». 


168 


FUNDAMENTAL  EQUATIONS 


If  the  vertex  M  moves  along  a  curve  other  than  a  parametric 
line,  that  is,  along  a  curve  determined  by  a  value  of  dv/du,  the 
variations  of  a,  £,  c  are  evidently  given  by 


da  du      da  dv 
du  ds      dv  ds 


^  ^_ 

du  ds      dv  ds 


do  du      do  dv 
du  ds      dv  ds 


in  which  the  differential  quotients  have  the  above  values. 

69.  Fundamental  equations  of  condition.  Suppose  that  we  asso 
ciate  with  the  trihedral  T  a  second  trihedral  TQ  whose  vertex  0  is 
fixed  in  space,  about  which  it  revolves  in  such  a  manner  that  its 
edges  are  always  parallel  to  the  corresponding  edges  of  T,  as  the 
vertex  of  the  latter  moves  over  the  surface  in  a  given  manner. 
The  position  of  T0  is  completely  determined  by  the  nine  direction- 
cosines  of  its  edges  with  three  mutually  perpendicular  lines  Lv  L2, 
Ls  through  0.  Call  these  direction-cosines  av  b{,  cl  ;  a2,  52,  c 


3, 


<?3.    These  functions  must  satisfy  the  equations 


(47) 


da      , 
-  =  6r- 

^w 

—  =  br  - 

dv~''   TI 


If  we  equate  the 


db 

—  =  cp  —  ar, 

du 

dc,                , 
^  =  «?-ip, 

db  _ 

dv 

dc                 , 
-  =  aq-lPl. 

-irolnQO      f\f     . 

rkVvf.o  i  ma  rl     fvrvm 

cucv 


two  of  these  equations,  and  in  the  reduction  of  the  resulting  equa 
tion  make  use  of  (47),  we  find 


Since  this  equation  must  be  true  when  b  and  c  have  the  values 
b  ,  <?  ;  5  ,  £0;  63,  <?8,  the  expressions  in  parenthesis  must  be  equal 

**  ^>O  T  /^9 


to  zero.    Proceeding  in  the  same  manner  with  - 
obtain  the  following  fundamental  equations  *  : 

dp      dpi 


and 


d*c 


dudv 


»  we 


(48) 


dq      dql 

dr      drl 


*  These  equations  were  first  obtained  by  Combescure,  Annales  de  VEcole  Normale, 
Ser.  1,  Vol.  IV  (1867),  p.  108;  cf.  also  Darboux,  Vol.  I,  p.  48. 


ROTATIONS  169 

These  necessary  conditions  upon  the  six  functions  p,  -  •  - ,  r1?  in 
order  that  the  nine  functions  ax,  •  •  • ,  cs  may  determine  the  position 
of  the  trihedral  T0,  are  also  sufficient  conditions.  The  proof  of  this 
is  similar  to  that  given  in  §  65. 

Equations  (47)  have  been  obtained  by  Darboux  *  from  a  study 
of  the  motion  of  the  trihedral  TQ.  He  has  called  jt?,  q,  •  •  • »  rl  the 
rotations. 

We  return  to  the  consideration  of  the  moving  trihedrals  T  and 
tu.  Let  (x,  y,  z)  and  (#',  ?/,  z')  denote  the  coordinates  of  a  point  P 
with  respect  to  T  and  tu  respectively.  Between  these  coordinates 
the  following  relations  hold : 

(x  =  xf  cos  U  —  (y'  sin  o>M  —  z'  cos  WM)  sin  £7, 
«/  =  #'  sin  CT  +  (#'  sin  Wu  —  2'  cos  &>M)  cos  £/, 
z  —  y1  cos  <WM  +  2'  sin  o>M. 

If  in  a  displacement  of  P  absolute  increments  with  respect  to 
the  trihedral  tu  at  M  be  indicated  by  S,  and  increments  relative  to 
these  moving  axes  by  c?,  we  have,  from  §  16, 

^L^-S^+i,     ¥  =  ^_'  +  -  +  -,     ^.  =  dzL-y-. 

d«u      dsu      pu  dsu      dsu      pu      TM'         C?SM      dsu      ru 

From  (49),  (50),  and  (45)  we  obtain  the  following! : 

$x      dx        r— 

—  =  —  4-  V E cos  U—ry  +  qz, 
du      du 

-^  =  —  +  VfismU—pz  +  ra, 

^W        ^M 

§2      dz 

—  = ox  +  PV. 

aw    a% 

Equations  similar  to  these  follow  also  from  the  consideration 
of  the  trihedral  tv.  Hence,  when  the  trihedral  T  moves  over  the 
surface  with  its  vertex  M  describing  a  curve  determined  by  a 
value  of  dv/du,  the  increments  of  the  coordinates  of  a  point 
P(x,  y,  z),  in  the  directions  of  the  axes  of  the  trihedral,  in  the 

*  L.c.,  Vol.  I,  chaps,  i  and  v. 

t  In  deriving  these  equations  we  have  made  use  of  the  fact  that  equations  (49)  define 
a  transformation  of  coordinates,  and  consequently  hold  when  the  coordinates  are  replaced 
by  the  projections  of  an  absolute  displacement  of  P. 


170  FUNDAMENTAL  EQUATIONS 

absolute  displacement  of  P,  which  may  also  be  moving  relative 
to  these  axes,  have  the  values  * 


where  we  have  put 


The  coordinates  of  M  are  (0,  0,  0),  so  that  the  increments  of  its 
displacements  are 
(53)        Sx  =  1;  du  +  ^dv,         §y  =  vidu  +  'rildv,         Sz  =  0. 

If  fa,  yv  2J  denote  the  coordinates  of  M  with  respect  to  the 
fixed  axes  formed  by  the  lines  Lv  £2,  L3  previously  defined,  it 
follows  that 


2,    2 


and  similar  expressions  for  yl  and  z^  where  at,  61?  ^;  «2,  62 
ag,  68,  c3  are  the  direction-cosines  of  the  fixed  axes  with  reference 
to  the  moving  axes.  Since  the  latter  satisfy  equations  (47),  the 
conditions  that  the  two  values  of  —  ^-  obtained  from  (54)  be  equal, 


a2  dzz    cu 

and  similarly  for  —*£  and  --£-1  are 
J  dudv 


(55) 


When  we  have  ten  functions  f ,  f  p  ?/,  77^  p,  p^  <?,  q^  r,  rx,  satis 
fying  these  conditions  and  (48),  the  functions  a1?  •  •-,  cs  can  be 
found  by  the  solution  of  a  Riccati  equation,  and  x^  y#  zl  by  quad 
ratures.  Hence  equations  (48)  and  (55)  are  sufficient  as  well  as 
necessary,  and  consequently  are  equivalent  to  the  Gauss  and 
Codazzi  equations. 

*  Cf.  Darboux,  Vol.  II,  p.  348. 


LINES  OF  CURVATUKE  171 

70.  Linear  element.    Lines  of  curvature.    From  (53)  we  see  that 
the  linear  element  of  the  surface  is 

(56)  ds2  =  (%du  +  ^  dvf  +(ndu  +  r)i  dv}\ 

Hence  a  necessary  and  sufficient  condition  that  the  parametric 
lines  be  orthogonal  is 

(57)  ff!+^i==0. 

For  a  sphere  of  radius  c  the  coordinates  of  the  center  are  (0,  0,  -  c),  it  being 
assumed  that  the  positive  normal  is  directed  outwards.    As  this  is  a  fixed  point,  it 
follows  from  equations  (51)  that  whatever  be  the  value  of  dv/du  we  must  have 
£du  +  hdv  -  (qdu  +  qidv)c  =  0, 

ydu  +  171  dv  +  (pdu  +  pidv)c  =  0, 
and  consequently 

£        i)        £1        ^i 

/KQ\  —  zr  =  —  =  —   —  =:  C. 

q       -P      qi       -Pi 

Conversely,  when  these  equations  are  satisfied,  the  point  (0,  0,  —  c)  is  fixed  in  space, 
and  therefore  the  surface  is  a  sphere.  Moreover,  suppose  that  we  have  a  propor 
tion  such  as  (58),  where  the  factor  of  proportionality  is  not  necessarily  constant. 
For  the  moment  call  it  t.  When  the  values  from  (58)  are  substituted  in  (55)  and 
reduction  is  made  in  accordance  with  (48)  we  get 

dt  dt  3t          dt 

r?^"7?1^=°'        *to~*lto 

Hence  t  is  constant  unless  ^  -  f^  is  zero,  which,  from  (56)  and  §  31,  is  seen  to 
be  possible  only  in  case  the  surface  is  isotropic  developable. 

By  definition  (§51)  a  line  of  curvature  is  a  curve  along  which 

the  normals  to  the  surface  form  a  developable  surface.    When  the 

vertex  is  displaced  along  one  of  these  lines,  a  point  (0,  0,  p)  must 

move  in  such  a  way  that  Bx  and  %  are  zero.    Hence  we  must  have 

f  du  -h  ^dv  +  (qdu  +  qldv)p  =  0, 

r]du  +  i]1dv  —  (pdu+pldv)p  =  0. 

Eliminating  p  and  dv/du  respectively,  we  obtain  the  equation  of 
the  lines  of  curvature, 

(59)  (f  du  +  ^dv)  (p  du  +  pvdv)  +  (17  du  +  rj^v)  (q  du  +  q^v)  =  0, 
and  the  equation  of  the  principal  radii, 

(60)  pz  (pq,  -  qPl)  +  p  (qrj,  -  q,rj  +  p^  -  p£ )  +  (fa  -  ^)  =  0. 

From  (59)  it  follows  that  a  necessary  and  sufficient  condition 
that  the  parametric  lines  be  the  lines  of  curvature  is 

(61)  fe>  +  i#  =  0,         f^i+ih^O. 


172  FUNDAMENTAL  EQUATIONS 

We  may  replace  these  equations  by 

P  =  \rj,       ?  =  -xf,       ^^V?!,       ?i  =  -\fn 

thus  introducing  two  auxiliary  functions  X  and  \.    When  these 
values  are  substituted  in  the  third  of  (55),  we  have 


If  X  and  \  are  equal,  the  above  equations  are  of  the  form  (58), 
which  were  seen  to  be  characteristic  of  the  sphere  and  the  iso- 
tropic  developable.  Hence  the  second  factor  is  zero,  so  that  equa 
tions  (61)  may  be  replaced  by 

(62)  ffi+  ^=0,         m+??i=<>» 
or 

(63)  ^=17  =  0,  P  =  q1=Q- 

From  (52)  it  follows  that  in  the  latter  case  the  x-  and  ?/-axes  are 
tangent  to  the  curves  v  =  const,  and  u  =  const.  We  shall  consider 
this  case  later. 

From  (60)  and  (52)  we  find  that  the  expression  for  the  total 
curvature  of  the  surface  is 


where  co  denotes  the  angle  between  the  parametric  curves.    Hence 
the  third  of  equations  (48)  may  be  written 

/g4\  V7£6rsin  co       H        dr      dr. 


PiP*  PiP 


71.  Conjugate  directions  and  asymptotic  directions.  Spherical 
representation.  We  have  found  (§  54)  that  the  direction  in  the 
tangent  plane  conjugate  to  a  given  direction  is  the  characteristic 
of  this  plane  as  it  envelopes  the  surface  in  the  given  direction. 
Hence,  from  the  point  of  view  of  the  moving  trihedral,  the  direc 
tion  conjugate  to  a  displacement,  determined  by  a  value  of  dv/du, 
is  the  line  in  the  #?/-plane  which  passes  through  the  origin,  and 
which  does  not  experience  an  absolute  displacement  in  the 
direction  of  the  2;-axis.  From  the  third  of  equations  (51)  it  is 
seen  that  the  equation  of  this  line  is 

(65)  (p  du  H-  p^v)  y  —  (q  du  +  q^dv)  x  =  0. 


CONJUGATE  DIRECTIONS  173 

If  the  increments  of  u  and  v,  corresponding  to  a  displacement  in 
the  direction  of  this  line,  be  indicated  by  d^  and  d^v,  the  quan 
tities  x  and  y  are  proportional  to  (f  d^u  +  f  ^v)  and  (r;  d^  4-  rj^v). 
When  x  and  ?/  in  (65)  are  replaced  by  these  values,  the  resulting 
equation  may  be  reduced  to 

(66)      (prj  -  gf)  dudjU  +  (pr]l  -  qgj  dudy  +  (p&  -  qg)  d^udv 


In  consequence  of  (55)  the  coefficients  of  dud^v  and  d^udv  are 
equal,  so  that  the  equation  is  symmetrical  with  respect  to  the 
two  sets  of  differentials,  thus  establishing  the  fact  that  the  rela 
tion  between  a  line  and  its  conjugate  is  reciprocal. 

In  order  that  the  parametric  lines  be  conjugate,  equation  (66) 
must  be  satisfied  by  du  =  0  and  d^v  =  0.    Hence  we  must  have 

(67) 


It  should  be  noticed  that  equations  (61)  are  a  consequence  of  the 
first  of  (62)  and  (67).  Hence  we  have  the  result  that  the  lines  of 
curvature  form  the  only  orthogonal  conjugate  system. 

From  (66)  it  follows  that  the  asymptotic  directions  are  given  by 

(68)  (prj  -  gf  )  du*  +  (prjl  -  q^  +p^  -  q£)  dudv  +  (p^  -  q^)  dv2  =  0. 

The  spherical  representation  of  a  surface  is  traced  out  by  the 
point  m,  whose  coordinates  are  (0,  0,  1)  with  respect  to  the  tri 
hedral  T0  of  fixed  vertex.  From  (51)  we  find  that  the  projections 
of  a  displacement  of  m,  corresponding  to  a  displacement  along  the 
surface,  are 

(69)  SX=qdu  +  qldv,          &Y=  —  (pdu+pldv),          &£=  0. 
Hence  the  linear  element  of  the  spherical  representation  is 

(70)  da2  =  (qdu  +  q^v)2  +  (pdu+  p^dv)\ 

The  line  defined  by  (65)  is  evidently  perpendicular  to  the  direc 
tion  of  the  displacement  of  m,  as  given  by  (69).  Hence  the  tangent 
to  the  spherical  representation  of  a  curve  upon  a  surface  is  perpen 
dicular  to  the  direction  conjugate  to  the  curve  at  the  corresponding 
point.  Therefore  the  tangents  to  a  line  of  curvature  and  its  rep 
resentation  are  parallel,  whereas  an  asymptotic  direction  and  its 
representation  are  perpendicular  (§  61). 


174  FUNDAMENTAL  EQUATIONS 

72.  Fundamental  relations  and  formulas.    From  equations  (53) 
and  (69)  we  have,  for  the  point  M  on  the  surface, 

—  =  £        ^  =         —  =  0 
du      *'       ° 

(71) 

""  =  ?,      —  =i?i,     —  =  0; 
and 

5v  ~V  £  \7~  ^\  7 

cu  du  du 


(72) 


•§.«. 


Consequently  the  following  relations  hold  between  the  fundamental 
coefficients,  the  rotations,  and  the  translations: 
F=  f  £  +  ^,      G  =  f  • 

(73) 


When,  in  particular,  the  parametric  system  on  a  surface  is  orthog 
onal,  and  the  x-  and  y-  axes  of  the  trihedral  are  tangent  to  the  curves 
v  =  const,  and  u  =  const,  through  the  vertex,  equations  (52)  are 

(74)  f=V5,       17  =  £=0, 

and  equations  (55)  reduce  to 
(76)    r—  L 


Moreover,  equations  (45)  and  the  similar  ones  for  plt  q^  rt  become 
(76) 


P'-      ""^'T.. 


. 

The  first  two  of  equations  (75)  lead,  by  means  of  (76),  to 

sin  w  1      d^/~E          shift\          1 

PU       "  VEG   fo  PV 

which  follow  also  from  §  58. 

The  third  of  equations  (75)  establishes  the  fact,  previously 
remarked  in  §  59,  that  the  geodesic  torsion  in  two  orthogonal 
directions  differs  only  in  sign. 


FUNDAMENTAL  RELATIONS 


175 


The  variations  of  the  direction-cosines  X\,  Y\,  Z\  of  the  tangent  to  the  curve 
u  —  const,  are  represented  by  the  motion  of  the  point  (1,  0,  0)  of  the  trihedral  T0 
with  fixed  vertex.  From  (51)  we  have 

5-5Ti  5^1  dZ\ 

du  cu  du 

(78) 


dv 


cv 


From  these  equations  we  see  that  as  a  point  describes  a  curve  v  =  const.  ,  namely 
CM,  the  tangent  to  this  curve  undergoes  an  infinitesimal  rotation  consisting  of  two 
components,  one  in  amount  rdu  about  the  normal  to  the  surface  and  the  other, 

—  qdu,  about  the  line  in  the  tangent  plane  perpendicular  to  the  tangent  to  Cu. 
Consequently,  by  their  definition,  the  geodesic  and  normal  curvature  of  Cu  are 
r/^/E  and  —  q/^/E  respectively.    Moreover,  it  is  seen  from  (72)  that  as  a  point 
describes  Cu  the  normal  to  the  surface  undergoes  a  rotation  consisting  of  the  com 
ponents  q  du  about  the  line  in  the  tangent  plane  perpendicular  to  the  tangent,  and 

—  p  du  about  the  tangent.    Hence,  if  Cu  were  a  geodesic,  the  torsion  would  be 
p/VE  to  within  the  sign  at  least.    Thus  by  geometrical  considerations  we  have 
obtained  the  fundamental  relations  (76). 

We  suppose  now  that  the  parametric  system  is  any  whatever. 
From  the  definition  of  the  differential  parameters  (§  37)  it  follows 

that 


E  = 


G  = 


Consequently  if  P,  $,  ^  denote  functions  similar  to  p,  q,  r,  for  a 


general  curve 


. 
v)  =  const. 


which  passes  through  M  and  whose  tangent  makes  the  angle 
with  the  moving  z-axis,  we  have,  from  (45), 


(79) 


T,  /^      1\  ,     •    ^  cos  ^1 
cos  <£  (  —  ---   +  sm  <I>  -   , 

\ds      r/  p 


P^VA^T 

, T  idco      1\ 

)  =  H.  V A.cf)    sin  4> —  cos  <£ 

\ds      rj 


cos  « 


P 


sn 


where  by  (III,  51)  ff~2  =  A 
any  other  family  of  curves 

Moreover,  equations  analogous  to  (44)  are 

db  __  cP  —  aR  dc 

ds 


—  AX2((£,  T|T)  and  ty  =  const,  defines 


da  _  bE  —  cQ 


aQ  — 


176  FUNDAMENTAL  EQUATIONS 

T.  .  da      da  du      da  dv 

If  now  in  -—  =  - — —  H — 

as      cu  as      dv  as 

we  replace  the  expressions  for  —  and  —  from  (47),  and  similarly 
for  db/ds  and  dc/ds,  we  obtain 

Pd8=Hl\/Al(f>(p  du  +p1dv),          Qds=H^/~K^>(qdu  +  q^dv), 
Eds  =  7^  VA~C/>  (r  du  -f  r^  dv). 

From  these  equations  and  (79)  we  derive  the  following  funda 
mental  formulas : 


(80) 


( — j  ds  =  cos  <&(p  du  +  pl  dv)  +  sin  <£  (q  du  +  ql  dv), 

\  as      T  I 

ds  =  sin  <I>  (p  du  +  p1  dv)  —  cos  4>  (<?  c?t*  -f  q^  dv), 

P 

sin  oj      d<&         du          dv 
l 


p          ds         ds          ds 

By  means  of  the  last  of  equations  (80)  we  shall  express  the 
geodesic  curvature  of  a  curve  in  terms  of  the  functions  E,  F,  G, 
of  their  derivatives,  and  of  the  angle  6  which  the  curve  makes 
with  the  curve  v  =  const.  If  we  take  the  rr-axis  of  the  trihedral 
tangent  to  the  curve  v  =  const.,  we  obtain  from  the  last  of  (80), 
in  consequence  of  (45), 

1       d0      ^/E  du      /V '  G      dco\dv 
Po~~  ds       Pg»   ds      \Pffv       dv/ds 
From  (III,  15,  16)  we  obtain 


dv       If    2  EG  \      dv  dv/      dv 

When  this  value  and  the  expressions  for  pgu  and  pgv  (IV,  57)  are 
substituted  in  the  above  equation,  we  have  the  formula  desired: 


-  L__  __ 

'  + 


2dvds      2H\du      E  dv    ds 


EXAMPLES 


1.  A  necessary  and  sufficient  condition  that  the  origin  of  the  trihedral  T  be  the 
only  point  in  the  moving  zy-plane  which  generates  a  surface  to  which  this  plane  is 
tangent,  is  that  the  surface  be  nondevelopable. 


PARALLEL  SURFACES  177 

2.  Determine  p  so  that  the  point  of  coordinates  (p,  0,  0)  with  respect  to  T  shall 
describe  a  surface  to  which  the  x-axis  of  T  is  normal ;  examine  the  case  when  the 
lines  of  curvature  are  parametric  and  the  x-axis  is  tangent  to  the  curve  v  =  const. 

3.  When  the  parametric  curves  are  minimal  lines  for  both  the  surface  and  the 
sphere,  it  is  necessary  that 

or  77  =  —  i£,        ifji  =  i£i,  q  =  ip,  q\  =  —  ipi\ 

in  this  case  the  parametric  curves  on  the  surface  form  a  conjugate  system,  and  the 
surface  is  minimal  (cf.  §  55). 

4.  When  the  asymptotic  lines  on  a  surface  form  an  orthogonal  system,  we 
must  have  ^  +  ^  =  0^        ^  +  ^  =  Q> 

in  which  case  the  surface  is  minimal. 

5.  When  the  lines  of  curvature  are  parametric,  and  the  x-axis  of  T  is  tangent 
to  the  curve  v  =  const.,  equations  (80)  reduce  to 

1      dw      /I        1\    .  cosw      cos2*      sin2* 

-j-  =  ( )  sin  *  cos  4>,         = H , 

T      as      \PI     PZ/  p  pi  PZ 

sin  w  _  d<£  1       /  q  dp\  du      p\  dPz  dv\ 

p         ds      Pz  —  Pi  \pi  cv  ds       q   du  ds) 

6.  When  the  second  equation  in  Ex.  5  is  differentiated  with  respect  to  s,  the 
resulting  equation  is  reducible  to 

cos  u  dp      sinw/   dw      2\  _    2  dp\  /du\s          2dPi/du\2dv 
P2     ds        p     \   ds      T/  du  \ds/  dv  \ds/  ds 

„  dpo  du  /du\2        „  dp»  /dv\8 


7.  On  a  surface  a  given  curve  makes  the  angle  *  with  the  x-axis  of  a  trihedral  T; 
the  point  P0  of  coordinates  cos  <t>,  sin  <J>,  0  with  reference  to  the  parallel  trihedral  TO 
with  fixed  vertex,  describes  the  spherical  indicatrix  of  the  tangent  to  the  curve ; 
the  direction-cosines  of  the  tangent  to  this  curve  are 

—  sin  *  sin  w,         cos  <£  sin  w,        cos  w, 

where  w  has  the  significance  indicated  in  §  49,  and  the  linear  element  is  ds/p;  derive 
therefrom  by  means  of  (51)  the  second  and  third  of  formulas  (80). 

8.  The  point  #,  whose  coordinates  with  reference  to  T0  of  Ex.  7  are 

sin  *  cos  w,         —  cos  $  cos  w,        sin  w, 

describes  the  spherical  indicatrix  of  the  binormal  to  the  given  curve  on  the  surface, 
and  its  linear  element  is  ds/r;  derive  therefrom  the  first  of  formulas  (80). 

73.  Parallel  surfaces.  We  inquire  under  what  conditions  the 
normals  to  a  surface  are  normal  to  a  second  surface.  In  order  that 
this  be  possible,  there  must  exist  a  function  t  such  that  the  point 
of  coordinates  (0,  0,  Q,  with  reference  to  the  trihedral  7",  describes 
a  surface  to  which  the  moving  2-axis  is  constantly  normal.  Hence 


178  FUNDAMENTAL  EQUATIONS 

we  must  have  8z  =  0,  and  consequently,  by  equations  (51),  t  must 
be  a  constant,  which  may  have  any  value  whatever.  We  have, 
therefore,  the  theorem : 

If  segments  of  constant  length  be  laid  off  upon  the  normals  to  a  sur 
face,  these  segments  being  measured  from  the  surface,  the  locus  of  their 
other  end  points  is  a  surface  with  the  same  normals  as  the  given  surface. 

These  two  surfaces  are  said  to  be  parallel.  Evidently  there  is 
an  infinity  of  surfaces  parallel  to  a  given  surface,  and  all  of  them 
have  the  same  spherical  representation. 

Consider  the  surface  for  which  t  has  the  value  a,  and  call  it  $. 
From  (51)  it  follows  that  the  projections  on  the  axes  of  T  of  a  dis 
placement  on  S  have  the  values 

r  —  f  du  +  ^dv  -f  (q  du  +  q^dv)  a, 


(82) 

=  77  du  -f-  jj^dv  —  (p  du  +  Pidv)  a. 


Comparing  these  results  with  (53),  we  see  that  the  displacements 
on  the  two  surfaces  corresponding  to  the  same  value  of  dv/du  are 
parallel  only  in  case  equation  (59)  is  satisfied,  that  is,  when  the 
point  describes  a  line  of  curvature  on  S.  But  from  a  characteristic 
property  of  lines  of  curvature  (§  51)  it  follows  that  the  lines  of  curva 
ture  on  the  two  surfaces  correspond.  Hence  we  have  the  theorem  : 

The  tangents  to  corresponding  lines  of  curvature  of  two  parallel 
surfaces  at  corresponding  points  are  parallel. 

From  (82)  and  (73)  we  have  the  following  expressions  for  the 
first  fundamental  quantities  of  /SY:  y 


or,  in  consequence  of  (IV,  78), 


(84) 


/  PI   p 

i 

/ 


PARALLEL  SURFACES  179 

The  moving  trihedral  for  S  can  be  taken  parallel  to  T  for  £, 
and  thus  the  rotations  are  the  same  for  both  trihedrals ;  and  from 
(82)  it  follows  that  the  translations  have  the  values 

£  =  £  +  00,       li=£i+fl?i>       ^  =  i?-op,        *?i=>?i-api- 
On  substituting  in  the  equations  for  £  analogous  to  (59),  (60),  (66), 
we  obtain  the  fundamental  equations  for  S  in  terms  of  the  functions 
for  S.    Also  from  (73)  we  have  the  following  expressions  for  the 
second  fundamental  coefficients  for  S: 

(85)  D  =  D-a€,         D'  =  D'-a&,          D"  =  D"  -  ag. 

Since  the  centers  of  principal  curvature  of  a  surface  and  its 
parallel  at  corresponding  points  are  the  same,  it  follows  that 

(86)  Pi  =  Pi+a>         P2  =  P2+a' 

Suppose  that  we  have  a  surface  whose  total  curvature  is  constant 
and  equal  to  1/c2.  Evidently  a  sphere  of  radius  c  is  of  this  kind, 
but  later  (Chapter  VIII)  it  will  be  shown  that  there  is  a  large  group 
of  surfaces  with  this  property.  We  call  them  spherical  surfaces. 

From  (86)  we  have     ^  _  ^  (^  _  a)  =  ^ 

so  that  if  we  take  a  =  ±  c,  we  obtain 

I+l-±i. 

Pi     P*          c 
Hence  we  have  the  theorem  of  Bonnet :  * 

With  every  surface  of  constant  total  curvature  1/c2  there  are  asso 
ciated  two  surfaces  of  mean  curvature  ±  1/ey  they  are  parallel  to  the 
former  and  at  the  distances  :p  c  from  it. 

And  conversely, 

With  every  surface  whose  mean  curvature  is  constant  and  different 
from  zero  there  are  associated  two  parallel  surfaces,  one  of  which  has 
constant  total  curvature  and  the  other  constant  mean  curvature. 

74.  Surfaces  of  center.  As  a  point  M  moves  over  a  surface  S 
the  corresponding  centers  of  principal  curvature  Ml  and  Mz  describe 
two  surfaces  S1  and  S2,  which  are  called  the  surfaces  of  center  of  S. 
Let  Cl  and  (72  be  the  lines  of  curvature  of  S  through  M,  and  Dl  and 
7>2  the  developable  surfaces  formed  by  the  normals  to  S  along  Cl 

*Nouvelles  annales  de  mathematiques,  Ser.  1,  Vol.  XII  (1853),  p.  433. 


180 


FUNDAMENTAL  EQUATIONS 


and  C2  respectively.  The  edge  of  regression  of  Dv  denoted  by  I\, 
is  a  curve  on  Sl  (see  fig.  17),  and  consequently  Sl  is  the  locus  of 
one  set  of  evolutes  of  the  curves  Cl  on  S.  Similarly  $2  is  the  locus 
of  a  set  of  evolutes  of  the  curves  Cz  on  S.  For  this  reason  S1  and 
$2  are  said  to  constitute  the  evolute  of  S,  and  S  is  their  involute. 
Evidently  any  surface  parallel  to  S  is  also  an  involute  of  Sl  and  S2. 
The  line  M^M^  as  a  generator  of  Dv  is  tangent  to  I\  at  Mv 
and,  as  a  generator  of  D2,  it  is  tangent  to  F2  at  Mz.  Hence  it  is  a 

common  tangent  of  the  surfaces  Sl  and  Sz. 
From  this  it  follows  that  the  developable 
surface  D1  meets  S{  along  Tl  and  envelopes 
Sz  along  a  curve  F2.  Its  generators  are  con 
sequently  tangent  to  the  curves  conjugate 
to  Fg  (§  54).  In  particular,  the  generator 
-flfjJfg  is  tangent  to  F2,  and  therefore  the 
directions  of  F2  and  T2  at  Jf2  are  conjugate. 
Similar  results  follow  from  the  considera 
tion  of  Z>2.  Hence : 

On  the  surfaces  of  center  of  a  surface  S 
the  curves  corresponding  to  the  lines  of  cur 
vature  of  S  form  a  conjugate  system. 


FIG.  17 


Since  the  developable  D1  envelopes  *Sf2, 
the  tangent  plane  to  $0  at  M2  is  the  tangent 
plane  to  Dl  at  this  point.  But  the  tangent 
plane  at  M2  is  tangent  to  Dl  all  along  M1MZ  (§  25),  and  consequently 
it  is  determined  by  M^MZ  and  the  tangent  to  C[  'at  M.  Hence  the 
normal  to  S  at  M2  is  parallel  to  the  tangent  to  C2  at  M.  In  like 
manner,  the  normal  to  Sl  at  Ml  is  parallel  to  the  tangent  to  Cl  at  M. 
Thus,  through  each  normal  to  S  we  have  two  perpendicular  planes, 
of  which  one  is  tangent  to  one  surface  of  center  and  the  other  to 
the  second  surface.  But  each  of  these  planes  is  at  the  same  time 
tangent  to  one  of  the  developables,  and  is  the  osculating  plane  of 
its  edge  of  regression.  Hence,  at  every  point  of  one  of  these  curves, 
the  osculating  plane  is  perpendicular  to  the  tangent  plane  to  the 
sheet  of  the  evolute  upon  which  it  lies,  and  so  we  have  the  theorem : 
The  edges  of  regression  of  the  developable  surfaces  formed  by  the 
normals  to  a  surface  along  the  lines  of  curvature  of  one  family  are 


SURFACES*  OF  CENTER  181 

geodesies  on  the  surface  of  center  which  is  the  locus  of  these  edges  ; 
and  the  developable  surf  aces  formed  ly  the  normals  along  the  lines  of 
curvature  in  the  other  family  envelope  this  surface  of  center  along  the 
curves  conjugate  to  these  geodesies. 

In  the  following  sections  we  shall  obtain,  in  an  analytical  manner, 
the  results  just  deduced  geometrically. 

75.  Fundamental  quantities  for  surfaces  of  center.  As  the  trihe 
dral  T  moves  over  the  surface  S  the  point  (0,  0,  p^  describes  the 
surface  of  center  Sr  Let  the  lines  of  curvature  on  S  be  parametric, 
and  the  z-axis  of  T  be  tangent,  to  the  curve  v  —  const.  Now 

\  /  '  &  L  J.  J.  L  '  ±  f^  -I.     A  f\ 

ri  rz 

so  that  the  first  two  of  equations  (48)  may  be  put  in  the  form 

JI, ___,,=,_    -,_    __.' 

(88) 


The  projections  on  the  moving  axes  of  the  absolute  displace 
ment  of  J/J  corresponding  to  a  displacement  of  M  on  S  are  found 
from  (51)  to  be 

(89)  Bxl  =  0,     S^  =  (rjl  —  p^pj  dv  =  V6r  ( 1 )dv,     Szj  =  dpr 

Hence  the  linear  element  of  Sl  is 

/      p  V 

(90)  ds*=  dri.+  Q(I-^]dfi 


consequently  the  curves  p^=  const,  on  Sl  are  the  orthogonal  tra 
jectories  of  the  curves  v  =  const.,  which  are  the  edges  of  regression, 
I\,  of  the  developables  of  the  normals  to  S  along  the  lines  of 
curvature  v  =  const. 

Let  us  consider  the  moving  trihedral  T^  for  Sl  formed  by  the 
tangents  to  the  curves  v  =  const,  and  pl  —  const,  at  M^  and  the  nor 
mal  at  this  point.  From  (89)  it  follows  that  the  first  tangent  has 
the  same  direction  and  sense  as  the  normal  to  S,  and  that  the  sec 
ond  tangent  has  the  same  direction  as  the  tangent  to  u  =  const,  on 
S,  the  sense  being  the  same  or  different  according  as  (1—  pl/p2)  is 


182  FUNDAMENTAL  EQUATIONS 

positive  or  negative.  And  the  normal  to  Sl  has  the  same  direction 
as  the  tangent  to  v  =  const,  on  £,  and  the  contrary  or  same  sense 
accordingly. 

If  then  we  indicate  with  an  accent  quantities  referring  to  the 
moving  trihedral  Tv  we  have 


(a'=c,  l'=€bj  c'=  —  ea, 

where  e  is  ±1  according  as  (1  —  pjp^  is  positive  or  negative.    From 
(89)  it  follows  that 


(92) 


When  the  values  (91)  are  substituted  in  equations  for  2\  similar 
to  equations  (47),  we  find 


Since  /  is  zero,  it  follows  from  (76)  that  the  curves  v  =  const,  are 
geodesies,  as  found  geometrically. 

The  various  fundamental  equations  for  St  may  now  be  obtained 
by  substituting  these  values  in  the  corresponding  equations  of  the 
preceding  sections.  Thus,  from  (73)  we  have 


which  follow  likewise  from  (90);  and  also 


Hence    the   parametric    curves   on   Sl  form   a  conjugate  system 
(cf.  §  54). 

The  equation  of  the  lines  of  curvature  may  be  written 


and  the  equation  of  the  asymptotic  directions  is 

^^-41^=0. 

p?  du  pl$u 


SURFACES  OF  CENTER  183 

The  expression  for  K^  the  total  curvature  of  S^  is 

(98)  ^-—L-Jj. 

~du 

From  (80)  and  (93)  it  follows  that  the  geodesic  curvature  at  Ml  of 
the  curve  on  Sl  which  makes  the  angle  <&l  with  the  curve  v  =  const, 
through  M^  is  given  by 


Hence  the  radius  of  geodesic  curvature  of  a  curve  pl  =  const.,  that 
is,  a  curve  for  which  <J>t  is  a  right  angle,  has,  in  consequence  of 
(87),  the  value  pl  —  p0.  In  accordance  with  §  57  the  center  of  geo 
desic  curvature  is  found  by  measuring  off  the  distance  pl  —  />2,  in  the 
negative  direction,  on  the  2-axis  of  the  trihedral  T.  Consequently 
Mz  is  this  center  of  curvature.  Hence  we  have  the  following  theo 
rem  of  Beltrami: 

The  centers  of  geodesic  curvature  of  the  curves  p^  =  const,  on  St 
and  of  p2  =  const,  on  S.,  are  the  corresponding  points  on  $2  and  Sl 
respectively. 

For  the  sheet  $2  of  the  evolute  we  find  the  following  results  : 


(90')  d**  =  E\-        du2  + 

the  equation  of  the  lines  of  curvature  is 
(96')   r**^ 
the  equation  of  the  asymptotic  lines  is 

^          5£*'-BS"' 

the  expression  for  the  total  curvature  is 

8ft 

*"5FS-5 

dv 


184  FUNDAMENTAL  EQUATIONS 

In  consequence  of  these  results  we  are  led  to  the  following 
theorems  of  Ribaucour,*  the  proof  of  which  we  leave  to  the  reader  : 

A  necessary  and  sufficient  condition  that  the  lines  of  curvature  upon 
Sj  and  S2  correspond  is  that  pl  —  p2=  c  (a  constant);  then  K^  —  K^ 
=  —  1/c2,  and  the  asymptotic  lines  upon  S1  and  $2  correspond. 

A  necessary  and  sufficient  condition  that  the  asymptotic  lines  on  Sl 
and  S2  correspond  is  that  there  exist  a  functional  relation  between  p^ 
and  p2. 

76.  Surfaces  complementary  to  a  given  surface.  We  have  just 
seen  that  the  normals  to  a  surface  are  tangent  to  a  family  of  geo 
desies  on  each  surface  of  centers.  Now  we  prove  the  converse  : 

The  tangents  to  a  family  of  geodesies  on  a  surface  Sl  are  normal 
to  an  infinity  of  parallel  surfaces. 

Let  the  geodesies  and  their  orthogonal  trajectories  be  taken  for 
the  curves  v  —  const,  and  u  =  const,  respectively,  and  the  param 
eters  chosen  so  that  the  linear  element  has  the  form 


We  refer  the  surface  to  the  trihedral  formed  by  the  tangents  to 
the  parametric  curves  and  the  normal,  the  z-axis  being  tangent  to 
the  curve  v  =  const.  Upon  the  latter  we  lay  off  from  the  point  Ml 
of  the  surface  a  length  X,  and  let  P  denote  the  other  extremity. 
As  M1  moves  over  the  surface  the  projections  of  the  corresponding 
displacements  of  P  have  the  values 


(99)          d\  +  du,        VX  +  X  ~l  dv,      -  X  (y,du  +  q,dv). 

In  order  that  the  locus  of  P  be  normal  to  the  lines  J^P,  we 
must  have  d\  +  du  =  0,  and  consequently 

X  =  —  u  +  £, 

where  c  denotes  the  constant  of  integration  whose  value  determines 
a  particular  one  of  the  family  of  parallel  surfaces.  If  the  direction- 
cosines  of  M^P  with  reference  to  fixed  axes  be  Xv  Yv  Z^  the 
coordinates  of  the  surface  /S,  for  which  c  =  0,  are  given  by 


where  x^  y^  zl  are  the  coordinates  of  Mr 

*  Comptes  Rendus,  Vol.  LXXIV  (1872),  p.  1399. 


COMPLEMENTAKY  SUKFACES  185 

The  surface  S1  is  one  of  the  surfaces  of  center  of  S.  In  order 
to  find  the  other,  $2,  we  must  determine  X  so  that  the  locus  of  P 
is  tangent  at  P  to  the  zz-plane  of  the  moving  trihedral.  The  con 
dition  for  this  is 


Hence  S2  is  given  by 

^LV          ^i 

y<i — y\       i —  •*!>    ^2  —  ^i       / — 
aV^  gVg. 

tfM  dw  dM 

and  the  principal  radii  of  S  are  expressed  by 

(10°)  Pl  =  u,        Pz=u- 

du 

Bianchi*  calls  S2  the  surface  complementary  to  Sl  for  the  given 
geodesic  system. 

Beltrami  has  suggested  the  following  geometrical  proof  of  the 
above  theorem.  Of  the  involutes  of  the  geodesies  v  —  const,  we 
consider  the  single  infinity  which  meet  S^  in  one  of  the  orthogonal 
trajectories  u  =  UQ.  We  shall  prove  that  the  locus  of  these  curves 
is  a  surface  S,  normal  to  the  tangents  to  the  geodesies.  Consider 
the  tangents  to  the  geodesies  at  the  points  of  meeting  of  the  latter 
with  a  second  orthogonal  trajectory  u  =  ur  The  segments  of  these 
tangents  between  the  points  of  contact  M  and  the  points  P  of 
meeting  with  S  are  equal  to  one  another,  because  they  are  equal 
to  the  length  of  the  geodesies  between  the  curves  u  —  UQ  and  u  =  ur 
Hence,  as  M  moves  along  an  orthogonal  trajectory  u  =  ul  of  the 
lines  JfP,  P  describes  a  second  orthogonal  trajectory  of  the  latter. 
Moreover,  as  M  moves  along  a  geodesic,  P  describes  an  involute 
which  is  necessarily  orthogonal  to  MP.  Since  two  directions  on  S 
are  perpendicular  to  JfP,  the  latter  is  normal  to  S. 

EXAMPLES 

1.  Obtain  the  results  of  §  73  concerning  parallel  surfaces  without  making  use  of 
the  moving  trihedral. 

2.  Show  that  the  surfaces  parallel  to  a  surface  of  revolution  are  surfaces  of 
revolution. 

*Vol.  I,  p.  293. 


186  FUNDAMENTAL  EQUATIONS 

3.  Determine  the  conjugate  systems  upon  a  surface  such  that  the  corresponding 
curves  on  a  parallel  surface  form  a  conjugate  system. 

4.  Determine  the  character  of  a  surface  S  such  that  its  asymptotic  lines  corre 
spond  to  conjugate  lines  upon  a  parallel  surface,  and  find  the  latter  surface. 

5.  Show  that  when  the  parametric  curves  are  the  lines  of  curvature  of  a  surface, 
the  characteristics  of  the  7/z-plane  and  zz-plane  respectively  of  the  moving  trihe 
dral  whose  x-axis  is  tangent  to  the  curve  v  =  const,  at  the  point  are  given  by 

(r  du  +  ri  dv)  y  —  q  (z  —  pi)  du  =  0, 
(r  du  +  r\  dv)  x  —  pi(z  —  p%)  dv  =  0  ; 

and  show  that  these  equations  give  the  directions  on  the  surfaces  Si  and  S2  which 
are  conjugate  to  the  direction  determined  by  dv/du. 

6.  Show  that  for  a  canal  surface  (§  29)  one  surface  of  centers  is  the  curve  of 
centers  of  the  spheres  and  the  other  is  the  polar  developable  of  this  curve. 

7.  The  surfaces  of  center  of  a  helicoid  are  helicoids  of  the  same  axis  and 
parameter  as  the  given  surface. 

GENERAL  EXAMPLES 

1.  If  t  is  an  integrating  factor  of   ^Edu-\  --  -    —  dv,  and  t0  the  conjugate 

v^ 

imaginary  function,  then  A2log  V#0  is  equal  to  the  total  curvature  of  the  quadratic 
form  E  du2  +  2  Fdudv  +  Gdv2,  all  the  functions  in  the  latter  being  real. 

2.  Show  that  the  sphere  is  the  only  real  surface  such  that  its  first  and  second 
fundamental  quadratic  forms  can  be  the  second  and  first  forms  respectively  of 
another  surface. 

3.  Show  that  there  exists  a  surface  referred  to  its  lines  of  curvature  with  the 
linear  element  ds2  =  eau(du*  +  du2),  where  a  is  a  constant,  and  that  the  surface  is 
developable. 

4.  When  a  minimal  surface  is  referred  to  its  minimal  lines 


hence  the  lines  of  curvature  and  asymptotic  lines  can  be  found  by  quadratures. 

5.  Establish  the  following  identities  in  which  the  differential  parameters  are 
formed  with  respect  to  the  linear  element  : 


. 

6.  Prove  that  (cf.  Ex.  2,  p.  1G6) 

A2*  =  -  4k  —I-  +  -}-  —  -(-  +  -}-  x(l  +  -V 

VJE^PI    f*      -VGCV\PI    f»)       \PI     PZ/ 


GENEEAL  EXAMPLES  187 

7.  Show  that  z2  +  ?/2  +  z2  =  "FT2  +  Ai  TF, 

the  differential  parameter  being  formed  with  respect  to  (23). 

8.  A  necessary  and  sufficient  condition  that  all  the  curves  of  an  orthogonal 
system  on  a  surface  be  geodesies  is  that  the  surface  be  developable. 

9.  If  the  geodesic  curvature  of  the  curves  of  an  orthogonal  system  is  constant 
(different  from  zero)  all  over  the  surface,  the  latter  is  a  surface  of  constant  negative 
curvature. 

10.  When  the  linear  element  of  a  surface  is  in  the  form 

ds2  =  du2  +  2  cos  u  dudv  +  dto2, 

the  parametric  curves  are  said  to  form  an  equidistantial  system.    Show  that  in  this 
case  the  coordinates  of  the  surface  are  integrals  of  the  system 


du  dv  du  dv  du  dv 


dy  dz      dz  dy      cz  dx      dz  dx      ex  dy  _  dx  dy_ 
cu  dv      du  dv      cu  dv      dv  cu      cu  dv      dv  cu 

11.  If  the  curves  v  =  const.,  u  =  const,  form  an  equidistantial  system,  the  tan 
gents  to  the  curves  v  =  const,  are  orthogonal  to  the  lines  joining  the  centers  of  geo 
desic  curvature  of  the  curves  u  =  const,  and  of  their  orthogonal  trajectories. 

12.  Of  all  the  displacements  of  a  trihedral  T  corresponding  to  a  small  displace 
ment  of  its  vertex  M  over  the  surface  there  are  two  which  reduce  to  rotations ;  they 
occur  when  M  describes  either  of  the  lines  of  curvature  through  the  point,  and  the 
axes  of  rotation  are  situated  in  the  planes  perpendicular  to  the  lines  of  curvature, 
each  axis  passing  through  one  of  the  centers  of  principal  curvature. 

13.  When  a  surface  is  referred  to  its  lines  of  curvature,  the  curves  defined  by 


a2  irl  dM3  +  3  g2  —  duzdv  +  3p?  —  dudv2  +  P?  —  dvs  =  0 
du  dv  du  dv 

possess  the  property  that  the  normal  sections  in  these  directions  at  a  point  are 
straight  lines,  or  are  superosculated  by  their  circles  of  curvature  (cf .  Ex.  9,  p.  21 ; 
Ex.  6,  p.  177).  These  curves  are  called  the  superosculating  lines  of  the  surface. 

14.  Show  that  the  superosculating  lines  on  a  surface  and  on  a  parallel  surface 
correspond. 

15.  Show  that  the  Gauss  equation  (64)  can  be  put  in  the  following  form  due  to 
Liouville : 


pgu  du\  pgv  )      du  dv 

where  pgu  and  pgv  denote  the  radii  of  geodesic  curvature  of  the  curves  v  =  const,  and 
u  =  const,  respectively. 

16.  When  the  parametric  curves  form  an  orthogonal  system,  the  equation  of 
Ex.  15  may  be  written 


_!\_J:  ___  L 

pgv)    p%u    P%V 


VE  du  \ 

17.  Determine  the  surfaces  which  are  such  that  one  of  them  and  a  parallel 
divide  harmonically  the  segment  between  the  centers  of  principal  curvature. 


188  FUNDAMENTAL  EQUATIONS 

18.  Determine  the  surfaces  which  are  such  that  one  of  them  and  a  parallel 
admit  of  an  equivalent  representation  (cf.  Ex.  14,  p.  113)  with  lines  of  curvature 

.  corresponding. 

19.  Derive  the  following  properties  of  the  surface 

a2  _  ft2     uv  Va2  -  62  v  V&2  —  w2          _  Va2  —  &2  u  Vu2  —  a2  _ 

ab      u  +  u '  b  M  +  U  a  w  +  v 

(i)  the  parametric  lines  are  plane  lines  of  curvature ; 
(ii)  the  principal  radii  of  curvature  are  p\  =  i>,  p%  =  —  u ; 
(iii)  the  surface  is  algebraic  of  the  fourth  order ; 
(iv)  the  surfaces  of  center  are  focal  conies. 

20.  Given  a  curve  C  upon  a  surface  S  and  the  ruled  surface  formed  by  the  tan 
gents  to  S  which  are  perpendicular  to  C  at  its  points  M ;  the  point  of  each  generator 
M. N  at  which  the  tangent  plane  to  the  ruled  surface  is  perpendicular  to  the  tan 
gent  plane  at  M  to  S  is  the  center  of  geodesic  curvature  of  C  at  M ;  when  the  ruled 
surface  is  developable,  this  center  of  geodesic  curvature  is  the  point  of  contact  of 
MN  with  the  edge  of  regression. 

21.  If  two  surfaces  have  the  same  spherical  representation  of  their  lines  of 
curvature,  the  locus  of  the  point  dividing  the  join  of  corresponding  points  in  con 
stant  ratio  is  a  surface  with  the  same  representation. 

22.  The  locus  of  the  centers  of  geodesic  curvature  of  a  line  of  curvature  is  an 
evolute  of  the  latter. 

23.  Show  that  when  E,  F,  G  ;  D,  IX,  IX'  of  a  surface  are  functions  of  a  single 
parameter,  the  surface  is  a  helicoid,  or  a  surface  of  revolution. 


CHAPTER  VI 

SYSTEMS  OF  CURVES.    GEODESICS 

77.  Asymptotic  lines.  We  have  said  that  the  asymptotic  lines 
on  a  surface  are  the  double  family  of  curves  whose  tangents  at 
any  point  are  determined  in  direction  by  the  differential  equation 

D  du2  +  2  D'dudv  +  D"dv2  =  0. 

These  directions  are  imaginary  and  distinct  at  an  elliptic  point, 
real  and  distinct  at  a  hyperbolic  point,  and  real  and  coincident  at  a 
parabolic  point.  If  we  exclude  the  latter  points  from  our  discussion, 
the  asymptotic  lines  may  be  taken  for  parametric  curves.  A  neces 
sary  and  sufficient  condition  that  they  be  parametric  is  (§55) 
(1)  D  =  Dn=Q. 

Then  from  (IV,  25)  we  have 

_  _D^_      !_ 

where  p  as  thus  denned  is  called  the  radius  of  total  curvature. 
The  Codazzi  equations  (V,  13')  may  be  written 


of  which  the  condition  of  integrability  is 

a  ri2i     d  ri2i 
<4>  ail  i  -hail  2  r 

In  consequence  of  (V,  3)  this  is  equivalent  to 


In  §  64  we  saw  that  K  is  a  function  of  E,  F,  G  and  their  deriva 
tives.  Hence  equations  (3)  are  two  conditions  upon  the  coefficients 
of  a  quadratic  form 

(6)  E  du2  +  2  Fdudv  +  G  dv2, 

189 


190  SYSTEMS  OF  CURVES 

that  it  may  be  the  linear  element  of  a  surface  referred  to  its  asymp 
totic  lines.  When  these  conditions  are  satisfied  the  function  D'  is 
given  by  (2)  to  within  sign.  Hence,  if  we  make  no  distinction  be 
tween  a  surface  and  its  symmetric  with  respect  to  a  point,  from  §  65 
follows  the  theorem : 

A  necessary  and  sufficient  condition  that  a  quadratic  form  (6)  be 
the  linear  element  of  a  surface  referred  to  its  asymptotic  lines  is  that 
its  coefficients  satisfy  equations  (3);  when  they  are  satisfied,  the  surface 
is  unique. 

For  example,  suppose  that  the  total  curvature  of  the  surface  is  the  same  at 
every  point,  thus  j 

a2 
where  a  is  a  constant.    In  this  case  equations  (3)  are 

cv          cu  cv  du 

which,  since  H2  ^  0,  are  equivalent  to 

dv  du 

Hence  E  is  a  function  of  u  alone,  and  G  a  function  of  v  alone.  By  a  suitable  choice 
of  the  parameters  these  two  functions  may  be  given  the  value  a2,  so  that  the  linear 
element  of  the  surface  can  be  written 

(7)  ds2  =  a2  (du2  +  2  cos  o>  dudv  +  dv2), 

where  w  denotes  the  angle  between  the  asymptotic  lines.  Thus  far  the  Codazzi  equa 
tions  are  satisfied  and  only  the  Gauss  equation  (V,  12)  remains  to  be  considered. 
When  the  above  values  are  substituted,  this  becomes 

(8)  —  —  sinw. 

dudv 

Hence  to  every  solution  of  this  equation  there  corresponds  a  surface  of  constant 

curvature whose  linear  element  is  given  by  (7). 

a2 

The  equation  of  the  lines  of  curvature  is  du2  —  dv2  =  0,  so  that  if  we  put 
u  -f  v  —  2  M!,  u  —  v  =  2  «!,  the  quantities  u\  and  v\  are  parameters  of  the  lines  of  cur 
vature,  and  in  terms  of  these  the  equation  of  the  asymptotic  lines  is  du}  —  dv}  =  0. 
Hence,  when  either  the  asymptotic  lines  or  the  lines  of  curvature  are  known  upon 
a  surface  of  constant  curvature,  the  other  system  can  be  found  by  quadratures. 

When  the  asymptotic  lines  are  parametric,  the  Gauss  equations 

(V,  7)  may  be  written 

•^  +  ^  +  5^1  =  0, 
/OX  i  du          du         dv 

(y) 

'dv72      aifru    '    ldv~'    ' 


ASYMPTOTIC  LINES 


191 


where  a,  5,  ax,  b1  are  determinate  functions  of  u  and  v,   and  in 

consequence  of  (5) 

(10) 


da 

Jo 


du 


Conversely,  if  two  such  equations  admit  three  real  linearly  inde 
pendent  integrals  f^u,  v),  fz(u,  v),  f3(u,  v),  the  equations 

<*  =/l(w»  V),  #  =/2(M,  V),  ^  =/>(!*,  1') 

define  a  surface  on  which  the  parametric  curves  are  the  asymptotic 
lines.  For,  by  the  elimination  of  a,  6,  a^  bl  from  the  six  equations 
obtained  by  replacing  6  in  (9)  by  x,  y,  z  we  get 


d2x 

tfy 

dzz 

du* 

du* 

£i? 

dx 

tji 

dz 

du 

du 

du 

dx 

dy_ 

dz 

'dv 

dv 

dv  . 

n  ju         uy         v*>  f\ 

7  7  7          =  "» 


=  0, 


which  are  equivalent  to  (1),  in  consequence  of  (IV,  2,  5).* 
As  an  example,  consider  the  equations 


dv2 

s 

dv2 

dx 

ty 

dz 

du 

du 

du 

dx 

dy 

dz 

dv 

dv 

dv 

of  which  the  general  integral  is  auv  +  bu  +  cv  +  d,  where  a,  b,  c,  rf  are  constants. 
By  choosing  the  fixed  axes  suitably,  the  most  general  form  of  the  equations  of  the 
surface  may  be  put  in  the  form 

From  these  equations  it  is  seen  that  all  the  asymptotic  lines  are  straight  lines,  so 
that  the  surface  is  a  quadric.  Moreover,  by  the  elimination  of  u  and  v  from  these 
equations  we  have  an  equation  of  the  form  z  =  ax-  -\-  2hxy  +  by2  +  ex  +  dy.  Hence 
the  surface  is  a  paraboloid. 

78.  Spherical  representation  of  asymptotic  lines.  From  (IV,  77) 
we  have  that  the  total  curvature  of  a  surface,  referred  to  its  asymp 
totic  lines,  may  be  expressed  in  the  form 

(ii)  A'=-^' 

where  ff-"  =  (o£—  o^2,  the  linear  element  of  the  spherical  represen- 


tation  being 


da'2  =  (odu2  +  2  &dudv  + 


*  Darbonx,  Vol.  I,  p.  138.  It  should  be  noticed  that  the  above  result  shows  that  the 
condition  that  equations  (9)  admit  three  independent  integrals  carries  with  it  not  only 
(10)  but  all  other  conditions  of  integrability. 


192  SYSTEMS  OF  CURVES 

From  this  result  and  (2)  it  follows  that  * 

5! 

#' 

Hence  the  fundamental  relations  (IV,  74)  reduce  to 

/-«  o\  Jf  —  n^/""         •        Jf _  r?  "-£ 

and  equations  (V,  26)  may  be  written 

'"  A^  ^X          P  /  **dX         e$X\  3X__^__  _ 

/{  \         du  cv 

Moreover,  the  Codazzi  equations  (V,  27)  are  reducible  to 


Consider  now  the  converse  problem  : 

To  determine  the  condition  to  be  satisfied  by  a  parametric  system 
of  lines  on  the  sphere  in  order  that  they  may  serve  as  the  spherical 
representation  of  the  asymptotic  lines  on  a  surface. 

First  of  all,  equations  (15)  must  satisfy  the  condition  of  integra- 
bility.  Then  p  is  obtainable  by  a  quadrature.  The  corresponding 
values  of  x,  y,  z  found  from  equations  (14)  and  from  similar  ones 
are  the  coordinates  of  a  surface  upon  which  the  asymptotic  lines 
are  parametric.  For,  it  follows  from  (14)  that 


du  **  dv  dv 

Furthermore,  p  is  determined  to  within  a  constant  factor  ;  conse 
quently  the  same  is  true  of  x,y,z\  therefore  the  surface  is  unique 
to  within  homothetic  transformations.  Hence  we  have  the  following 
theorem  of  Dini  : 

A  necessary  and  sufficient  condition  that  a  double  family  of  curves 
upon  the  sphere  be  the  spherical  representation  of  the  asymptotic  lines 
upon  a  surface  is  that  &,  &,  $  satisfy  the  equation 

V    d  ri2 


the  corresponding  surfaces  are  homothetic  transforms  of  one  another, 
and  their  Cartesian  coordinates  are  found  by  quadratures. 

*  The  choice  P  =  —  D'/ft  gives  the  surface  symmetric  to  the  one  corresponding  to  (12), 
as  is  seen  from  (14),  and  hence  may  be  neglected. 


FORMULAS  OF  LELIEUVKE 


193 


When  equations  (1)  obtain,  the  fundamental  equations  (V,  28) 
lead  to  the  identities 

|in  =  r  11  v   2  ri2V    |22 1  =  r 22V_  r  ri2i ' 
121       ri2v  ri2i       ri2v 


(18) 


i 


rm 

l2/=    - 

rny  r22j      py 

\2J'  llJ-     llJ 


The  third  and  fourth  of  these  equations  are  consequences  also  of 
(3)  and  (15). 

79.  Formulas   of  Lelieuvre.    Tangential  equations.     In   conse 
quence  of  (V,  31)  equations  (14)  may  be  put  in  the  form 


where  e  is  ±  1  according  as  the  curvature  of  the  surface  is  positive 
or  negative.    Hence,  if  we  put 

(20)  ^  =  V-€/)X,          v2  =  V^epY,          vs  =  ^/ 

we  have  the  following  formulas  due  to  Lelieuvre  :  * 


'dx  _ 

du      du 

dx  _ 
dv 

fa<2         fa* 

dv      dv 

(21) 

dy  _ 
^w 

du      du 

dv 

~dv      dv~ 

a^  _ 

du 

Vl            V2 

du      du 

dz  _ 

Vl          V* 

fai     fa% 

dv      dv 

The  conditions  of  integrability  of  these  equations  arc 

du  dv  _  du  dv  _  dudv 
~         v*  v* 

Bulletin  des  Sciences  Mathtmatiques,  Vol.  XII  (1888),  p.  126. 


194  ^  SYSTEMS  OF  CURVES 


By  means  of  (V,  22)  and  (15)  we  find  from  (20)  that  the  common 

ratio  of  these  equations  i 

,    ,.  .  ,  ,  ,. 

solutions  of  the  equation 


ratio  of  these  equations  is  —=  ^—  --  <&   Consequently  i^,  i/2,  vs  are 


dudv      \^/p  dudv 

Conversely,  we  have  the  theorem : 

G-iven  three  particular  integrals  v^  i>2,  vs  of  an  equation  of  the  form 

d20 
(22)  -^-  =  X0, 

where  \  is  any  function  whatever  of  u  and  v  ;  the  surface,  whose  co 
ordinates  are  given  by  the  corresponding  quadratures  (21),  has  the 
parametric  curves  for  asymptotic  lines,  and  the  total  curvature  of  the 
surface  is  measured  by 

/93\  K  ~ • — •  —  • • 

For,  from  (21),  it  is  readily  seen  that  v^  i/2,  vz  are  proportional  to 
the  direction-cosines  of  the  normal  to  the  surface.  And  if  these 
direction-cosines  be  given  by  (20),  we  are  brought  to  (19),  from 
which  we  see  that  the  conditions  (16)  are  satisfied. 

Take,  for  example,  the  simplest  case  - — —  =  0,  and  three  solutions 

j    /    \    i     /    /    \  //;        1     9     Q\ 

V{  —  0|  (U)  -(-  Yi(V).  (I  =  *j  &,  &) 

The  coordinates  of  the  surface  are 

/r 
j 

and  similar  expressions  for  y  and  z.    When,  in  particular,  we  take 

0,-  (u)  =  a,-w  +  &£,        $i  (v)  =  a  to  +  /S,-, 

the  expressions  for  x,  y,  z  are  of  the  form  auv  +  bu  +  cv  +  d,  and  consequently  the 
surface  is  a  paraboloid. 

From  equations  (V,  22,  34)  it  follows  that  when  the  asymptotic 
lines  are  parametric,  the  tangential  coordinates  X,  Y,  Z,  W  are 
solutions  of  the  equations 

HVd0 

18^-llJ  du 
\     '  I  ^2/j       ^22") ' 30 


CONJUGATE  SYSTEMS  195 

EXAMPLES 

1.  Upon  a  nondevelopable  surface  straight  lines  are  the  only  plane  asymptotic 
lines. 

2.  The  asymptotic  lines  on  a  minimal  surface  form  an  orthogonal  isothermal 
system,  and  their  spherical  images  also  form  such  a  system. 

3.  Show  that  of  all  the  surfaces  with  the  linear  element  ds2  =  du*  +  (u2  +  a2)  du2, 
one  has  the  parametric  curves  for  asymptotic  lines  and  another  for  lines  of  curva 
ture.    Determine  these  two  surfaces. 

4.  The  normals  to  a  ruled  surface  along  a  generator  are  parallel  to  a  plane. 
Prove  conversely,  by  means  of  the  formulas  of  Lelieuvre,  that  if  the  normals  to  a 
surface  along  the  asymptotic  lines  in  one  system  are  parallel  to  a  plane,  which 
differs  with  the  curve,  the  surface  is  ruled. 

5.  If  we  take  v^  =  u,  vz  =  D,  j>3  =  0(u),  the  formulas  of  Lelieuvre  define  the 
most  general  right  conoid. 

6.  If  the  asymptotic  lines  in  one  system  on  a  surface  be  represented  on  the 
sphere  by  great  circles,  the  surface  is  ruled. 

80.  Conjugate  systems  of  parametric  lines.  Inversions.  It  is  our 
purpose  now  to  consider  the  case  where  the  parametric  lines  of  a 
surface  form  a  conjugate  system.  As  thus  defined,  the  character 
istics  of  the  tangent  plane,  as  it  envelops  the  surface  along  a  curve 
v  =  const.,  are  the  tangents  to  the  curves  u  =  const,  at  their  points 
of  intersection  with  the  former  curve ;  and  similarly  for  a  plane 
enveloping  along  a  curve  u  =  const. 

The  analytical  condition  that  the  parametric  lines  form  a  conju 
gate  system  is  (§  54) 

(25)  D'=0. 

It  follows  immediately  from  equations  (V,  7)  that  x,  y,  z  are  solu 
tions  of  an  equation  of  the  type 

(26)  J^  +  a^  +  6^0, 

cudv         du         dv 

where  a  and  b  are  functions  of  u  and  v,  or  constants.    By  a  method 
similar  to  that  of  §  77  we  prove  the  converse  theorem : 

Iffi(u,  v),/2(M,  v),/3(w,  v)  be  three  linearly  independent  real  solu 
tions  of  an  equation  of  the  type  (26),  the  equations 

(27)  *  =  />,*),          y=fz(u,v),          *=ft(u,v) 

define  a  surface  upon  which  the  parametric  curves  form  a  conjugate 
system.* 

9  *  Cf .  Darboux,  Vol.  I,  p.  122. 


196  SYSTEMS  OF  CURVES 

We  have  seen  that  the  lines  of  curvature  form  the  only  orthog 
onal  conjugate  system.  Hence,  in  order  that  the  parametric  lines 
on  the  surface  (27)  be  lines  of  curvature,  we  must  have 

F^fa  +  tyty+tete^^ 

du  dv      du  dv      du  dv 

But  this  is  equivalent  to  the  condition  that  xz+yz+z2  also  be  a 
solution  of  equation  (26),  as  is  seen  by  substitution.  Hence  we 
have  the  theorem  of  Darboux  *  : 

If  x,  y,  z,  #2-{-  ?/2-f-  z*  are  particular  solutions  of  an  equation  of  the 
form  (26),  the  first  three  serve  for  the  rectangular  coordinates  of  a 
surface,  upon  which  the  parametric  lines  are  the  lines  of  curvature. 

Darboux  f  has  applied  this  result  to  the  proof  of  the  following 
theorem  : 

When  a  surface  is  transformed  ly  an  inversion  into  a  second  sur 
face,  the  lines  of  curvature  of  the  former  become  lines  of  curvature 
of  the  latter. 

By  definition  an  inversion,  or  a  transformation  by  reciprocal 
radii,  is  given  by 

* 


where  c  denotes  a  constant.    From  these  equations  we  find  that 
(29)  (if  +  f  +  z*)  (x?  +  y  «  +  z,')  =  c', 

and  by  solving  for  x,  y,  z,  (,f 


— 

~  "  '      ' 


yt+*l          *?+**-»- »,' 

If,  now,  the  substitution  Q  _  _       °" 

«?+*•+•? 

be  effected  upon  equation  (26),  the  resulting  equation  in  or  will 
admit,  in  consequence  of  (29)  and  (30),  the  solutions  xv  y^  zv  c4, 
and  therefore  is  of  the  form 

(31) 


*  Vol.  I,  p.  136.  t  Vol.  I,  p.  207. 


SURFACES  OF  TRANSLATION         197 

Moreover,  equation  (26)  admits  unity  for  a  particular  solution, 
and  consequently  x*  +  yl  +  z?  is  a  solution  of  (31),  which  proves 
the  theorem. 

As  an  example,  we  consider  a  cone  of  revolution.  Its  lines  of  curvature  are  the 
elements  of  the  cone  and  the  circular  sections.  When  a  transformation  by  recip 
rocal  radii,  whose  pole  is  any  point,  is  applied  to  the  cone,  the  transform  S  has  two 
families  of  circles  for  its  lines  of  curvature,  in  consequence  of  the  above  theorem 
and  the  fact  that  circles  and  straight  lines,  not  through  the  pole,  are  transformed 
into  circles.  Moreover,  the  cone  is  the  envelope  of  a  family  of  spheres  whose  cen 
ters  lie  on  its  axis,  and  also  of  the  one-parameter  family  of  tangent  planes  ;  the 
latter  pass  through  the  vertex.  Since  tangency  is  preserved  in  this  transformation, 
the  surface  S  is  in  two  ways  the  envelope  of  a  family  of  spheres  :  all  the  spheres 
of  one  family  pass  through  a  point,  and  the  centers  of  the  spheres  of  the  other 
family  lie  in  the  plane  determined  by  the  axis  of  the  cone  and  the  pole. 

81.  Surfaces  of  translation.  The  simplest  form  of  equation  (26)  is 


dudv 

in  which  case  equations  (27)  are  of  the  type 
(32)  x  =  U1+r»         y=u^+V»         e  =  V>+r» 

where  U^  Z70,  Us  are  any  functions  whatever  of  u  alone,  and  V^  F2, 
F3  any  functions  of  v  alone.  This  surface  may  be  generated  by 
effecting  upon  the  curve 

Xl=UV  Vl=U»  21=^3 

a  translation  in  which  each  of  its  points  describes  a  curve  con 
gruent  with  the  curve 

*,  =  F,,         y,=  r,,         Z.2=F3. 

In  like  manner  it  may  be  generated  by  a  translation  of  the  second 
curve  in  which  each  of  its  points  describes  a  curve  congruent  with 
the  first  curve.  For  this  reason  the  surface  is  called  a  surface  of 
translation.  From  this  method  of  generation,  as  also  from  equa 
tions  (32),  it  follows  that  the  tangents  to  the  curves  of  one  family 
at  their  points  of  intersection  with  a  curve  of  the  second  family 
are  parallel  to  one  another.  Hence  we  have  the  theorem  of  Lie  *  : 

The  developable  enveloping  a  surface  of  translation  along  a  gener 
ating  curve  is  a  cylinder. 

*  Math.  Annalen,  Vol.  XIV  (1879),  pp.  332-367. 


198  SYSTEMS  OF  CURVES 

Lie  has  observed  that  the  surface  defined  by  (32)  is  the  locus  of 
the  mid-points  of  the  joins  of  points  on  the  curves 


It  may  be  that  these  two  sets  of  equations  define  the  same  curve 
in  terms  of  different  parameters.  In  this  case  the  surface  is  the 
locus  of  the  mid-points  of  all  chords  of  the  curve.  These  results 
are  only  a  particular  case  of  the  following  theorem,  whose  proof  is 
immediate  : 

The  locus  of  the  point  which  divides  in  constant  ratio  the  joins  of 
points  on  two  curves,  .or  all  the  chords  of  one  curve,  is  a  surface 
of  translation  ;  in  the  latter  case  the  curve  is  an  asymptotic  line  of 
the  surface. 

When  the  equations  of  a  surface  of  translation  are  of  the  form 
x=U,         y  =  V,         9=Ui+V» 

the  generators  are  plane  curves  whose  planes  are  perpendicular. 
We  leave  it  to  the  reader  to  show  that  in  this  case  the  asymptotic 
lines  can  be  found  by  quadratures. 

82.  Isothermal-conjugate  systems.  When  the  asymptotic  lines 
upon  a  surface  are  parametric,  the  second  quadratic  form  may  be 
written  X  dudv.  When  the  surface  is  real,  so  also  is  this  quadratic 
form.  Therefore,  according  as  the  curvature  of  the  surface  is  posi 
tive  or  negative,  the  parameters  u  and  v  are  conjugate-imaginary 
or  real.  r« 

We  consider  the  former  case  and  put 


when  u^  and  vl  are  real.  In  terms  of  these  parameters  the  second 
quadratic  form  is  \(du£+dvj).  Hence  the  curves  Mt=  const., 
vl  ==  const,  form  a  conjugate  system,  for  which 

(33)  D  =  D",         D'=0. 

Bianchi  *  has  called  a  system  of  this  sort  isothermal-conjugate.  Evi 
dently  such  a  system  bears  to  the  second  quadratic  form  an  ana 
lytical  relation  similar  to  that  of  an  isothermal-orthogonal  system 

*  Vol.  I,  p.  107. 


ISOTHERMAL-CONJUGATE  SYSTEMS  199 

to  the  first  quadratic  form.  In  the  latter  case  it  was  only  necessary 
that  EG—F*  be  positive,  and  the  analogous  requirement,  namely 
DD"  —  I)''2  >  0,  is  satisfied  by  surfaces  of  positive  curvature.  Hence 
all  the  theorems  for  isothermal-orthogonal  systems  (§§  40,  41)  are 
translated  into  theorems  concerning  isothermal-conjugate  systems 
by  substituting  Z>,  IX,  D"  for  E,  F,  G  respectively  in  the  formulas. 
In  particular,  we  remark  that  if  the  curves  u  =  const.,  v  =  const. 
on  a  surface  form  an  isothermal-conjugate  system,  all  other  real 
isothermal-conjugate  systems  are  given  by  u±  =  const.,  v  1  =  const., 
the  quantities  ul  and  vl  being  defined  by 

ui+  il\  —  (t)(u  i  w)» 
where  <£  is  any  analytic  function. 

When  the  curvature  of  the  surface  is  negative  and  we  put 


in  the  second  quadratic  form  \dudv,  it  becomes  \(du*—  dv*).  In 
this  case 

(34)  D  =  -!>",          D'=0. 

Hence  the  curves  w1  —  const,  and  i\  =  const,  form  a  conjugate  sys 
tem  which  may  be  called  isothermal-conjugate.  With  each  change 
of  the  parameters  u  and  v  of  the  asymptotic  lines  there  is  obtained 
a  new  isothermal-conjugate  system.  Hence  if  u  and  v  are  parame 
ters  of  an  isothermal-conjugate  system  upon  a  surface  of  negative 
curvature,  the  parameters  of  all  such  systems  are  given  by 


where  <£  and  ^r  denote  arbitrary  functions. 

It  is  evident  that  if  the  parameters  for  a  surface  are  such  that 

(35)  —  =  -,         Z/=0, 

D"      V 

where  U  and  V  are  functions  of  u  and  v  respectively,  then  by  a 
change  of  parameters  which  does  not  change  the  parametric  curves 
we  can  reduce  (35)  to  one  of  the  forms  (33)  or  (34).  Hence  equa 
tions  (35)  are  a  necessary  and  sufficient  condition  that  the  para 
metric  curves  form  an  isothermal-conjugate  system.  Referring  to 


200  SYSTEMS  OF  CUKVES 

§  77,  we  see  that  the  lines  of  curvature  upon  a  surface  of  constant 
total  curvature  form  an  isothermal-conjugate  system. 

When  equation  (35)  is  of  the  form  (33)  or  (34),  we  say  that  the 
parameters  u  and  v  are  isothermal-conjugate. 

83.  Spherical  representation  of  conjugate  systems.  When  the 
parametric  curves  are  conjugate,  equations  (IV,  69)  reduce  to 

-     GI?  FDD"  .,     EDm 

~-~W^  ~W  '-Ji- 

From  these  equations  and  (III,  15)  it  follows  that  the  angle  CD' 
between  the  parametric  curves  on  the  sphere  is  given  by 

,         &  F 

COS  o>  =  —  -  =  qp  —  -  ==  if  COS  O), 


where  the  upper  sign  corresponds  to  the  case  of  an  elliptic  point 
and  the  lower  to  a  hyperbolic  point.  Hence  we  have  the  theorem: 
The  angles  between  two  conjugate  directions  at  a  point  on  a  sur 
face,  and  between  the  corresponding  directions  on  the  sphere,  are  equal 
or  supplementary,  according  as  the  point  is  hyperbolic  or  elliptic. 

When   the    parametric    curves    form   a  conjugate  system,  the 
Codazzi  equations  (V,  27)  reduce  to 


and  equations  (V,  26)  become 

(dx       D  /          X         dX 


du          dv 

Hence,  when  a  system  of  curves  upon  the  sphere  is  given,  the 
problem  of  finding  the  surfaces  with  this  representation  of  a 
conjugate  system  reduces  to  the  solution  of  equations  (36)  and 
quadratures  of  the  form  (37),  after  X,  Y,  Z  have  been  determined 
by  the  solution  of  a  Riccati  equation.  By  the  elimination  of  D 
or  D"  from  equations  (36)  we  obtain  a  partial  differential  equation 
of  the  second  order. 


CONJUGATE  PARAMETRIC  SYSTEMS 


201 


From  the  general  equations  (V,  28)  we  derive  the  following, 
when  the  parametric  curves  form  a  conjugate  system  :  * 


(38) 


(fii\_8ioSD    rnv   /22\    aiogD"    /22V 
li/=  ~w  ~\ir  i2/=  ~fo~  ~i2/' 

/12\         D"  fllV  f!2\          D  /22V 

ilJ=    "T12/'  l2/=  ~I7'll/' 


22 


D"  f!2 


ll 


!2V 


84.  Tangential  coordinates.  Pro  jective  transformations.  The  prob 
lem  of  finding  the  surfaces  with  a  given  representation  of  a  con 
jugate  system  is  treated  more  readily  from  the  point  of  view  of 
tangential  coordinates.  For,  from  (V,  22)  and  (V,  34)  it  is  seen 
that  -3T,  r,  Z,  and  W  are  particular  solutions  of  the  equation 


<»»> 


Hence  every  solution  of  this  equation  linearly  independent  of 
A",  F,  Z  determines  a  surface  with  the  given  representation  of  a 
conjugate  system,  and  the  calculation  of  the  coordinates  2-,  y,  z 
does  not  involve  quadratures  (§  67). 

Conversely,  it  is  readily  seen  that  if  the  tangential  coordinates 
satisfy  an  equation  of  the  form 

d*e          30        00 

h  a  --  h  b  --  f-c#=0. 

du         dv 


the  coordinate  lines  form  a  conjugate  system  on  the  surface. 

As  an  example,  we  determine  the  surfaces  whose  lines  of  curvature  are  repre 
sented  on  the  sphere  by  a  family  of  curves  of  ccinstant  geodesic  curvature  and  their 
orthogonal  trajectories.  If  the  former  family  be  the  curves  v  =  const.,  and  if  the 
linear  element  on  the  sphere  be  written  da-2  —  Edu2  -f  Gdu2,  we  must  have  (IV,  60) 


where  0  (u)  is  a  function  of  v  alone.    By  a  change  of  the  parameter  v  this  may  be 
made  equal  to  unity.    In  this  case  equation  (39)  is  reducible  to 


du 


*Cf.  Bian«hi,  Vol.  I,  p.  167. 


202  SYSTEMS  OF  CUKVES 

The  general  integral  of  this  equation  is 


where  v0  denotes  a  constant  value  of  t>,  and  U  and  V  are  arbitrary  functions  of  u 
and  u  respectively.    Hence  : 

The  determination  of  all  the  surfaces  whose  lines  of  curvature  are  represented  on 
the  sphere  by  a  family  of  curves  of  constant  geodesic  curvature  and  their  orthogonal 
trajectories,  requires  two  quadratures. 

In  order  that  among  all  the  surfaces  with  the  same  represen 
tation  of  a  conjugate  system  there  may  be  a  surface  for  which  the 
system  is  isothermal-conjugate,  and  the  parameters  be  isothermal- 
conjugate,  it  is  necessary  that  equations  (36)  be  satisfied  by 
iX'ssiD,  according  as  the  total  curvature  is  positive  or  negative. 
In  this  case  equations  (36)  are 

01og.D_/12\'     fllV        alocrT)        12V       22V 


u 

The  condition  of  integrability  is 

a  rri2V    my-i     z  17121'   /22V  ] 

^Llim2)rdl2mi)J' 

When  this  is  satisfied  D  may  be  found  by  quadratures,  and  then 
the  coordinates,  by  (37).    Hence  we  have  the  theorem: 

A  necessary  and  sufficient  condition  that  a  family  of  curves  upon 
the  sphere  represent  an  isothermal-conjugate  system  on  a  surface, 
and  that  u  and  v  be  isothermal-conjugate  parameters,  is  that  £,  <^,  $ 
satisfy  (40);  then  the  surface  is  unique  to  within  its  homothetics, 
and  its  coordinates  are  given  by  quadratures.  ,-< 

The  following  theorem  concerning  the  in  variance  of  conjugate 
directions  and  asymptotic  lines  is  due  to  Darboux  : 

When  a  surface  is  subjected  to  a  protective  transformation  or  a 
transformation  by  reciprocal  polars,  conjugate  directions  and  asymp 
totic  lines  are  preserved. 

We  prove  this  theorem  geometrically.  Consider  a  curve  C  on  a 
surface  $and  the  developable  D  circumscribing  the  surface  along  C. 
When  a  projective  transformation  is  effected  upon  S  we  obtain  a 
surface  S19  corresponding  point  with  point  to  S,  and  C  goes  into  a 
curve  CjUpon  S^  and  D  in  to  a  developable  Dl  circumscribing  Sl  along 


PROJECTIVE  TRANSFORMATIONS  203 

Cj  ;  moreover,  the  tangents  to  C  and  (7X  correspond,  as  do  the  gener 
ators  of  D  and  Dr  Since  the  generators  are  in  each  case  tangent  to 
the  curves  conjugate  to  C  and  Cl  respectively,  the  theorem  is  proved. 
In  the  case  of  a  polar  reciprocal  transformation  a  plane  corre 
sponds  to  a  point  and  vice  versa,  in  such  a  way  that  a  plane  and  a 
point  of  it  go  into  a  point  and  a  plane  through  it.  Hence  S  goes 
into  Sv  C  into  D1,  D  into  Cv  and  the  tangents  to  C  and  generators 
of  D  into  the  generators  of  J\  and  tangents  to  <7r  Hence  the 
theorem  is  proved. 

EXAMPLES 

1.  Show  that  the  parametric  curves  on  the  surface 

-  ^I  +  FI  _  _C7A±Z2  -  u*  +  Fs 

:    U  +  V  '  -    u  +  V*  "1TTF" 

where  the  I7's  are  functions  of  u  alone  and  the  F's  of  v  alone,  form  a  conjugate  system. 

2.  On  the  surface  x  —  U\V\,  y  =  UZV\,  z  —  F2,  where  U\,  U2  are  functions  of 
u  alone  and  FI,  F2  of  v  alone,  the  parametric  curves  form  a  conjugate  system  and 
the  asymptotic  lines  can  be  found  by  quadratures. 

3.  The  generators  of  a  surface  of  translation  form  an  equidistantial  system 
(cf.  Ex.  10,  p.  187). 

4.  Show  that  a  paraboloid  is  a  surface  of  translation  in  more  than  one  way. 

5  .  The  locus  of  the  mid-points  of  the  chords  of  a  circular  helix  is  a  right  helicoid. 

Q.  Discuss  the  surface  of  translation  which  is  the  locus  of  points  dividing  in 
constant  ratio  the  chords  of  a  twisted  cubic. 

7.  From  (28)  it  follows  that 

_   2       ,   „       ,  a      c4  (dx2  +  dy2  +  dz2) 

dx?  +  dy?  +  dz?  =  —  —  : 

(x*  +  y*  +  2'2)2 

consequently  the  transformation  by  reciprocal  radii  is  conformal. 

8.  Determine  the  condition  to  be  satisfied  by  the  function  u  so  that  a  surface 
with  the  linear  element 


9/      <>,79, 
=  a?  (cos2  w  du2  + 

shall  have  the  total  curvature  —  I/a2.    Show  that  if  the  parametric  curves  are  the 
lines  of  curvature,  they  form  an  isothermal-conjugate  system. 

9.  A  necessary  and  sufficient  condition  that  the  linear  element  of  a  surface 
referred  to  a  conjugate  system  can  be  written 


is  that  the  parametric  curves  be  the  characteristic  lines.    Find  the  condition  imposed 
upon  the  curves  on  the  unit  sphere  in  order  that  they  may  represent  these  lines. 

10.  Conjugate  systems  and  asymptotic  lines  are  transformed  into  curves  of  the 
same  sort  when  a  surface  is  transformed  by  the  general  protective  transformation 

ABC 
X  =  D'        y  =  D'        *  =  D' 
where  A,  2>,  C,  D  are  linear  functions  of  the  new  coordinates  Xi,  y\,  z\. 


204  GEODESICS 

85.  Equations  of  geodesic  lines.  We  have  defined  a  geodesic  to 
be  a  curve  whose  geodesic  curvature  is  zero  at  every  point  ;  conse 
quently  its  osculating  plane  at  any  point  is  perpendicular  to  the 
tangent  plane  to  the  surface. 

From  (IV,  49)  it  follows  that  every  geodesic  upon  a  surface  is 
an  integral  curve  of  the  differential  equation 


(41)     , 


ds         ds/\     ds2          ds*/     \     ds 

.  /V—  +  7<T—  W—  --—  Y—  Y-4-  —  —  —  4  i—  f— 
\     ds         ds)l\du      2  to/\ds)      du  ds  ds      2  dv  \ds 

—\\-  —  (—\2-i-  —  —  —  ±fi—  —  -—\(—  Y  =  0 
ds/[2  du  \ds)      dv  ds  ds     \dv      2  du)\ds      ~ 


_/»•— 

\     ds 

If  the  fundamental  identity 


+2+ 

\ds/  ds  ds          \ds/ 

which  gives  the  relation  between  w,  v,  s  along  the  curve,  be  differ 
entiated  with  respect  to  s,  we  have 

du       d*u         d*v\         dv       d*u         d*v 


L+  2  4-24-  -4--= 

dv         du)\ds)  ds     \    dv      du)  ds  \dsj      dv 
If  this  equation  and  (41)  be  solved  with  respect  to 

/  72  72     \ 

,      T,tfu      ~d*v\  ,,   . 

and    F-—  +  G  -^  )»    we  obtain 
\     ds2         ds*/ 

F—  +  F—  4-  -  —  1^—  Y+  —  —  —  +(—  -  -  —  V—  V=  0 
ds2         ds2      2  du  \ds)      dv  ds  ds     \dv      2  du)\ds/ 


cFv     (W_lMy<faV     dGdu  dv      IdG/dv  \2=  Q 
dt     \9*      2  dv  )\ds)      du  ds  ds      2  Bv  \ds) 

If  these  equations  be  solved  with  respect  to  -—  and  —  ^  »  we  have, 
in  consequence  of  (V,  2), 


(42) 


d*u    riii/<fov   2i12\  — ^+{22V— Y=o 

111  /du\*  .  rt  fl21  (fw  dv  .  f 22- 


EQUATIONS  OF  GEODESICS  205 

Every  pair  of  solutions  of  these  equations  of  the  form  u  =/j(«), 
v  =/2(«),  determines  a  geodesic  on  the  surface,  and  s  is  its  arc. 

But  a  geodesic  may  be  defined  in  terms  of  u  and  v  alone,  without 
the  introduction  of  the  parameter  s.  If  v  =  <f>(u)  defines  such  a 
curve,  then  dv  du  d*v  /du\*  .,d*u 


ct  u 
Substituting  these   expressions  in  (42)  and  eliminating  —  >  we 

have,  to  within  the  factor  (du/ds)2, 
(43)  <//' 


From  (42)  it  follows  that  when  du/ds  is  zero, 


Hence,  when  this  condition  is  not  satisfied,  equation  (43)  defines 
the  geodesies  on  a  surface  ;  and  when  it  is  satisfied,  equations  (43) 
and  u  =  const,  define  them. 

From  the  theory  of  differential  equations  it  follows  that  there 
exists  a  unique  integral  of  (43)  which  takes  a  given  value  for 
u  —  MO,  and  whose  first  derivative  takes  a  given  value  for  u  =  UQ. 
Hence  we  have  the  fundamental  theorem  :  , 

Through  every  point  on  a  surface  there  passes  a  unique  geodesic 
with  a  given  direction. 

As  an  example,  we  consider  the  geodesies  on  a  surface  of  revolution.  We  have 
found  (§  46)  that  the  linear  element  of  such  a  surface  referred  to  its  meridians  and 
parallels  is  of  the  form 

(45)  ds2  =  (1  +  0'2)  du2  -f  wW, 

where  z  =  0  (u)  is  the  equation  of  the  meridian  curve.    If  we  put 

(46) 

and  indicate  the  inverse  of  this  equation  by  u  —  ^(wi),  we  have 

(47)  <Zs2  =  d^  +  fdw2, 

and  the  meridians  and  parallels  are  still  the  parametric  curves.    For  this  case  equa 

tions  (42)  are 

(48)          ^i  -  w*y  =  °-    —  +  —  —  -  =  °- 

(    '  <W       "  W  <W  ds  ds 


206  GEODESICS 

The  first  integral  of  the  second  is 

.do 

<//2  —  =  c, 
ds 

where  c  is  a  constant.    Eliminating  ds  from  this  equation  and  (47),  and  integrating, 
we  have 


(49)  c  f  - 

«/  , 


=  ±  w  + 


2  _  C2 

where  Ci  is  a  constant.    The  meridians  v  =  const,  correspond  to  the  case  c  =  0. 
Hence  we  have  the  theorem : 

The  geodesies  upon  a  surface  of  revolution  referred  to  its  meridians  and  parallels 
can  be  found  by  quadratures. 

It  should  be  remarked  that  equation  (49)  defines  the  geodesies  upon  any  surface 
applicable  to  a  surface  of  revolution. 

86.  Geodesic  parallels.  Geodesic  parameters.  From  (43)  it  fol 
lows  that  a  necessary  and  sufficient  condition  that  the  curves 
v  =  const,  on  a  surface  be  geodesies  is  that 


If  the  parametric  system  be  orthogonal,  this  condition  makes  it 
necessary  that  E  be  a  function  of  u  alone,  say  E  =  U2.  By  replacing 

I  U  du  by  u  we  do  not  change  the  parametric  lines,  and  E  becomes 
equal  to  unity.    And  the  linear  element  has  the  form 
(51)  d**=du*+Gdi?, 

where  in  general  G  is  a  function  of  both  u  and  v.  From  this  it 
follows  that  the  length  of  the  segment  of  a  curve  v  =  const,  between 
the  curves  u  =  UQ  and  u  =  u^  is  given  by 


/»«!  X»«! 

I    dsu=  I    du  =  u^—u^ 

«V«0  •'WO 


Since  this  length  is  independent  of  v,  it  follows  that  the  segments 
of  all  the  geodesies  v  =  const,  included  between  any  two  orthog 
onal  trajectories  are  of  equal  length.  In  consequence  of  the  funda 
mental  theorem,  we  have  that  there  is  a  unique  family  of  geodesies 
which  are  the  orthogonal  trajectories  of  a  given  curve  C.  The  above 
results  enable  us  to  state  the  following  theorem  of  Gauss  * : 

If  geodesies  be  drawn  orthogonal  to  a  curve  C,  and  equal  lengths  be 
measured  upon  them  from  C,  the  locus  of  their  ends  is  an  orthogonal 

trajectory  of  the  geodesies. 

*  L.C.,  p.  25. 


GEODESIC  PARALLELS  207 

This  gives  us  a  means  of  finding  all  the  orthogonal  trajectories 
of  a  family  of  geodesies,  when  one  of  them  is  known.  And  it  sug 
gests  the  name  geodesic  parallels  for  these  trajectories.  Referring 
to  §  37,  we  see  that  these  are  the  curves  there  called  parallels, 
and  so  the  theorem  of  §  37  may  be  stated  thus : 

A  necessary  and  sufficient  condition  that  the  curves  <f>  =  const,  be 
geodesic  parallels  is  that 

(52)  A,*  =/(*), 

where  the  differential  parameter  is  formed  with  respect  to  the  linear 
element  of  the  surface,  and  f  denotes  any  function.  In  order  that  <f> 
be  the  length  of  the  geodesic  curves  measured  from  the  curve  </>  =  0, 
it  is  necessary  and  sufficient  that 

(53)  A,*  =  l. 

Moreover,  we  have  seen  that  when  a  function  $  satisfies  (52),  a 
new  function  satisfying  (53)  can  be  found  by  quadrature.  When 
this  function  is  taken  as  u,  the  linear  element  has  the  form  (51). 
In  this  case  we  shall  call  u  and  v  geodesic  parameters. 

87.  Geodesic  polar  coordinates.  The  following  theorem,  due  to 
Gauss,*  suggests  an  important  system  of  geodesic  parameters: 

If  equal  lengths  be  laid  off  from  a  point  P  on  the  geodesies  through  P, 
the  locus  of  the  end  points  is  an  orthogonal  trajectory  of  the  geodesies. 

In  proving  the  theorem  we  take  the  geodesies  for  the  curves 
v  =  const.,  and  let  u  denote  distances  measured  along  these  geo 
desies  from  P.  The  points  of  a  curve  u  =  const,  are  consequently 
at  the  same  geodesic  distance  from  P,  and  so  we  call  them  geodesic 
circles.  It  is  our  problem  to  show  that  this  parametric  system  is 
orthogonal. 

From  the  choice  of  u  we  know  that  E  =  \,  and  hence  from  (50) 
it  follows  that  F  is  independent  of  u.  At  P,  that  is  for  u  =  0,  the 

derivatives  — »  — »  —  are  zero.    Consequently  F  and  G  are  zero 

Zv     dv    dv 

for  u  =  0,  and  the  former,  being  independent  of  w,  is  always  zero. 
Hence  the  theorem  is  proved. 

We  consider  such  a  system  and  two  points  MQ(u,  0),  J/^w,  vj 
on  the  geodesic  circle  of  radius  u.  The  length  of  the  arc  MQM1 

*L.c.,p.  24. 


208  GEODESICS 

is  given  by   /     ^Gdv.    As  u  approaches  zero  the  ratio 

Jo 

approaches  the  angle  between  the  tangents  at  P  to  the  geodesies 
v  =  0  and  v  =  v^    If  6  denotes  this  angle,  we  have 


a   .. 

6  =  lim  -  =  I       —  —       dv. 

u=o 


v    r 

=  I 

••/o 

In  order  that  v  be  6,  it  is  necessary  and  sufficient  that   -       =  1. 

L  du  Ju=0 

These  particular  geodesic  coordinates  are  similar  to  polar  coordi 
nates  in  the  plane,  and  for  this  reason  are  called  geodesic  polar 
coordinates.  The  above  results  may  now  be  stated  thus  : 

The  necessary  and  sufficient  conditions  that  a  system  of  geodesic 
coordinates  be  polar  are 


(54)  =0,  -i. 

L        J«=o  L  Bu   JM=0 

It  should  be  noticed,  however,  that  it  may  be  necessary  to  limit  the  part  of  the 
surface  under  consideration  in  order  that  there  be  a  one-to-one  correspondence 
between  a  point  and  a  pair  of  coordinates.  For,  it  may  happen  that  two  geodesies 
starting  from  P  meet  again,  in  which  case  the  second  point  of  meeting  would  be 
defined  by  two  sets  of  coordinates.*  For  example,  the  helices  are  geodesies  on  a 
cylinder  (§  12),  and  it  is  evident  that  any  number  of  them  can  be  made  to  pass 
through  two  points  at  a  finite  distance  from  one  another  by  varying  the  angle  under 
which  they  cut  the  elements  of  the  cylinder.  Hence,  in  using  a  system  of  geodesic 
polar  coordinates  with  pole  at  P,  we  consider  the  portion  of  the  surface  inclosed 
by  a  geodesic  circle  of  radius  r,  where  r  is  such  that  no  two  geodesies  through  P 
meet  within  the  circle,  t 

When  the  linear  element  is  in  the  form  (51),  the  equation  of 
Gauss  (V,  12)  reduces  to 

(55) 


If  KQ  denotes  the  total  curvature  of  the  surface  at  the  pole  P, 
which  by  hypothesis  is  not  a  parabolic  point,  from  (54)  and 
(55)  it  follows  that 


_0  _K 

o  '    L  <*     ~    °* 

*  Notice  that  the  pole  is  a  singular  point  for  such  a  system,  because  H*  =  0  for  u  =  0. 
tDarboux  (Vol.  II,  p.  408)  shows  that  such  a  function  r  exists;  this  is  suggested 
also  by  §  94. 


GEODESIC  POLAK  COOKDINATES  209 

Therefore,  for  sufficiently  small  values  of  w,  we  have 


. 

O 

Hence  the  circumference  and  area  of  a  geodesic  circle  of  radius  u 
have  the  values  * 


/*2 

=  I 

Jo 


2  ITU  - 


+  « 


„ 

where  et  and  e2  denote  terms  of  orders  higher  than  the  third  and 
fourth  respectively. 

EXAMPLES 

1.  Find  the  geodesies  of  an  ellipsoid  of  revolution. 

2.  The  equations  x  —  u,  y  =  v  define  a  representation  of  a  surface  with  the 
linear  element  ds2  =  v  (du2  +  dv2)  upon  the  xy-plane  in  such  a  way  that  geodesies 
on  the  former  are  represented  by  parabolas  on  the  latter. 

3.  Find  the  total  curvature  of  a  surface  with  the  linear  element 

(a2  -  v2)  du?  +  2  w>  dudv  +  (a2  -  u2)  dv* 

US      T   XV*  -  --  •  -  •  —  —  -   9 

(a2  _  W2  _  W2)2 

where  R  and  a  are  constants  and  integrate  the  equation  of  geodesies  for  the  surface. 

4.  A  twisted  curve  is  a  geodesic  on  its  rectifying  developable. 

5.  The  evolutes  of  a  twisted  curve  are  geodesies  on  its  polar  developable. 

6.  Along  a  geodesic  on  a  surface  of  revolution  the  product  of  the  radius  of  the 
parallel  through  a  point  and  the  sine  of  the  angle  of  inclination  of  the  geodesic 
with  the  meridian  is  constant. 

7.  Upon  a  surface  of  revolution  a  curve  cannot  be  a  geodesic  and  loxodromic 
at  the  same  time  unless  the  surface  be  cylindrical. 

8.  Upon  a  helicoid  the  orthogonal  trajectories  of  the  helices  are  geodesies  and 
the  other  geodesies  can  be  found  by  quadratures. 

9.  If  a  family  of  geodesies  and  their  orthogonal  trajectories  on  a  surface  form 
an  isothermal  system,  the  surface  is  applicable  to  a  surface  of  revolution. 

10.  The  radius  of  curvature  of  a  geodesic  on  a  cone  of  revolution  at  a  point  P 
varies  as  the  cube  of  the  distance  of  P  from  the  vertex. 

88.  Area  of  a  geodesic  triangle.  With  the  aid  of  geodesic  polar 
coordinates  Gauss  proved  the  following  important  theorem  f  : 

The  excess  over  180°  of  the  sum  of  the  angles  of  a  triangle  formed 
by  geodesies  on  a  surface  of  positive  curvature,  or  the  deficit  from  180° 

*  Bertrand,  Journal  de  Mathematiques,  Ser.  1,  Vol.  XIII  (1848),  pp.  80-86.      t  L.c.,  p.  30. 


210  GEODESICS 

of  the  sum  of  the  angles  of  such  a  triangle  on  a  surface  of  negative 
curvature,  is  measured  by  the  area  of  the  part  of  the  sphere  which 
represents  that  triangle. 

In  the  proof  of  this  theorem  Gauss  made  use  of  the  equation  of 
geodesic  lines  in  the  form 


where  6  denotes  the  angle  which  the  tangent  to  a  geodesic  at  a 
point  makes  with  the  curve  v  —  const,  through  the  point.  This 
equation  is  an  immediate  consequence  of  formula  (V,  81).  When 
the  parametric  system  is  polar  geodesic,  this  becomes 

(57)  M  =  -'-£*. 

Let  ABC  be  a  triangle  whose  sides  are  geodesies,  and  let  a,  /3,  7 
denote  the  included  angles.  From  (IV,  7  3)  it  follows  that  the  inclosed 
area  on  the  sphere  is  given  by 

(58)  d  =  f  f  //  dudv  =  €  Ipcff  dudv, 

where  e  is  ±  1  according  as  the  curvature  is  positive  or  negative, 
and  the  double  integrals  are  taken  over  the  respective  areas. 

Let  A  be  the  pole  of  a  polar  geodesic  system  and  AB  the  curve 
v  =  0.  From  (55)  and  (58)  we  have 


r«r 

] 

Jo  Jo 


, 

--  dvdu' 

o      o 


In  consequence  of  (54)  we  have,  upon  integration  with  respect  to  u, 


which,  by  (57),  is  equivalent  to 

&  =  e  f  dv  +  e  f    dd. 

Jo  Jn-ft 

For,  at  B  the  geodesic  BC  makes  the  angle  TT  —  fi  with  the  curve 
v  =  0,  and  at  C  it  makes  the  angle  7  with  the  curve  v  =  a.    Hence 

we  have  (7i  =  e(a  +  /3  +  7  -  TT), 

which  proves  the  theorem. 


AREA  OF  A  GEODESIC  TRIANGLE  211 

Because  of  the  form  of  the  second  part  of  (58)  Ci  may  be  said 
to  measure  the  total  curvature  of  the  geodesic  triangle,  so  that  the 
above  theorem  may  also  be  stated  thus  : 

The  total  curvature  of  a  geodesic  triangle  is  equal  to  the  excess 
over  180°,  or  deficit  from  180°,  of  the  sum  of  the  angles  of  the  tri 
angle,  according  as  the  curvature  is  positive  or  negative. 

The  extension  of  these  theorems  to  the  case  of  geodesic  polygons 
is  straightforward. 

In  the  preceding  discussion  it  has  been  tacitly  assumed  that  all  the  points  of  the 
triangle  ABC  can  be  uniquely  denned  by  polar  coordinates  with  pole  at  A.  We 
shall  show  that  this  theorem  is  true,  even  if  this  assumption  is 
not  made. 

If  the  theorem  is  not  true  for  ABC,  it  cannot  be  true  for 
both  of  the  triangles  ABD  and  ACD  obtained  by  joining  A 
and  the  middle  point  of  BC  with  a  geodesic  AD  (fig.  18).  For, 
by  adding  the  results  for  the  two  triangles,  we  should  have  the 
theorem  holding  for  ABC.  Suppose  that  it  is  not  true  for  A  BD. 
Divide  the  latter  into  two  triangles  and  apply  the  same  reason 
ing.  By  continuing  this  process  we  should  obtain  a  triangle  as 
small  as  we  please,  inside  of  which  a  polar  geodesic  system 
would  not  uniquely  determine  each  point.  But  a  domain  can  be  chosen  about  a 
point  so  that  a  unique  geodesic  passes  through  the  given  point  and  any  other  point 
of  the  domain.*  Consequently  the  above  theorem  is  perfectly  general. 

By  means  of  the  above  result  we  prove  the  theorem : 
Tivo  geodesies  on  a  surface  of  negative  curvature  cannot  meet  in 
two  points  and  inclose  a  simply  connected  area. 

Suppose  that  two  geodesies  through  a  point  A  pass  through  a 
second  point  B,  the  two  geodesies  inclosing  a  simply  connected 
portion  of  the  surface  (fig.  19).  Take  any  geodesic  cutting  these 
two  segments  AB  in  points  C  and  D.  Since 
the  four  angles  ACD,  ADC,  BCD,  BDC  are 
together  equal  to  four  right  angles,  the  sum 
/D  of  the  angles  of  the  two  triangles  ADC  arid 

BDC  exceed  four  right  angles  by  the  sum 
of  the  angles  at  A  and  B.    Therefore,  in 

consequence  of  the  above  theorem  of  Gauss,  the  total  curvature 

of  the  surface  cannot  be  negative  at  all  points  of  the  area  ADBC. 

On  the  contrary,  it  can  be  shown  that  for  a  surface  of  positive 

curvature  geodesies  through  a  point  meet  again  in  general.    In 

*  Darboux,  Vol.  II,  p.  408 ;  cf.  §  94. 


212  GEODESICS 

fact,  the  exceptional  points,  if  there  are  any,  lie  in  a  finite  portion 
of  the  surface,  which  may  consist  of  one  or  more  simply  connected 
parts.*  For  example,  the  geodesies  on  a  sphere  are  great  circles, 
and  all  of  these  through  a  point  pass  through  the  diametrically 
opposite  point.  Again,  the  helices  are  geodesies  on  a  cylinder 
(§  12),  and  it  is  evident  tnat  any  number  of  them  can  be  made  to 
pass  through  two  points  at  a  finite  distance  from  one  another  by 
varying  the  angle  under  A  hich  they  cut  the  elements  of  the  cyl 
inder.  Hence  the  domain  oi  a  system  of  polar  geodesic  coordinates 
is  restricted  on  a  surface  of  oositive  curvature. 

89.  Lines  of  shortest  length.  Geodesic  curvature.  We  are  now 
in  a  position  to  prove  the  theorem : 

If  two  points  on  a  surface  are  such  that  only  one  geodesic  passes 
through  them,  the  segment  of  the  geodesic  measures  the  shortest  dis 
tance  on  the  surface  between  the  two  points. 

Take  one  of  the  points  for  the  pole  of  a  polar  geodesic  system 
and  the  geodesic  for  the  curve  v  =  0.  The  coordinates  of  the 
second  point  are  (u^  0).  The  parametric  equation  of  any  other 
curve  through  the  two  points  is  of  the  form  v  =  ^>(w),  and  the 

length  of  its  arc  is 


f 

Jo 


Since  G  >  0,  the  value  of  this  in 
tegral  is  necessarily  greater  than 
Wj,  and  the  theprem  is  proved. 
By  means  of  equation  (57)  we  derive  another  definition  of  geo 
desic  curvature.  Consider  two  points  M  and  M'  upon  a  curve  C, 
and  the  unique  geodesies  g,  g'  tangent  to  C  at  these  points  (fig.  20). 
Let  P  denote  the  point  of  intersection  of  g  and  g\  and  Sty  the 
angle  under  which  they  cut.  Liouville  f  has  called  Sty  the  angle  of 
geodesic  contingence,  because  of  its  analogy  to  the  ordinary  angle 
of  contingence.  Now  we  shall  prove  the  theorem: 

The  limit  of  the  ratio  Sty/Ss,  as  M'  approaches  M,  is  the  geodesic 
curvature  of  C  at  M. 

*  For  a  proof  of  this  the  reader  is  referred  to  a  memoir  by  H.  v.  Mangoldt,  in  Crelle, 
Vol.  XCI  (1881),  pp.  23-53. 

t  Journal  de  Mathtmatiques,  Vol.  XVI  (1851),  p.  132. 


GEODESIC  ELLIPSES  AKD  HYPERBOLAS 

In  the  proof  of  this  theorem  we  take  for  parametric  curves  the 
given  curve  (7,  its  geodesic  parallels  and  their  geodesic  orthogonals, 
the  parameter  u  being  the  distance  measured  along  the  latter  from  C. 
Since  the  geodesic  g  meets  the  curve  v  =  v0  orthogonally,  the  angle 
under  which  it  meets  v  =  v'  may  be  denoted  by  ?r/2  4-  SO.  As  M ' 
approaches  Jf,  W  approaches  dd  given  by  y/>7),  and  the  sum  of  the 
angles  of  the  triangle  M'PQ  approaches  18C°.  Hence  S^/r  approaches 
—  dQ,  so  that  we  have 


Ss  ds 

»    v- 

which  is  the  expression  for  the  geodesic  curvature  of  the  curve  C. 
90.  Geodesic  ellipses  and  hyperbolas.  An  important  system  of 
parametric  lines  for  a  surface  is  formed  by  two  families  of  geodesic 
parallels.  Such  a  system  may  be  obtained  by  constructing  the  geo 
desic  parallels  of  two  curves  C^  and  (72,  which  are  not  themselves 
geodesic  parallels  of  one  another,  or  by  taking  the  two  families  of 
geodesic  circles  with  centers  at  any  two  points  F^  and  F2.  Let  u  and 
v  measure  the  geodesic  distances  from  C^  and  C2,  or  from  Fl  and  F2. 
They  must  be  solutions  of  (53).  Consequently,  in  terms  of  them, 
we  must  have  ^  Q. 

EG-F*~~  EG-F*~ 

If,  as  usual,  o>  denotes  the  angle  between  these  parametric  lines, 
we  have,  from  (III,  15,  16), 

v     n          *  T?     COSft) 

-U  =  U  =  — — —  >  JF  =     .  ? 

sin2  ft)  sin  <o 

so  that  the  linear  element  has  the  following  form,  due  to  Weingarten : 

/rrix  ,  9     du2+  2  cos  ft)  dudv  -f-  dv2 

(59)  ds*=-  .  a  -• 

sin2  &) 

Conversely,  when  the  linear  element  is  reducible  to  this  form, 
u  and  v  are  solutions  of  (53),  and  consequently  the  parametric 
curves  are  geodesic  parallels. 

In  terms  of  the  parameters  uv  and  v^  denned  by  u  =  i^-h  vl  and 
v  =  ul—  i\,  the  linear  element  (59)  has  the  form 

(60)  df^^aL  +  JuL. 


.        o  o 

sm          °°S 


2 


214  GEODESICS 

The  geometrical  significance  of  the  curves  of  parameter  i^  and  vl 
is  seen  when  the  above  equations  are  written 

The  curves  w1  =  const,  and  vl  =  const,  are  respectively  the  loci  of 
points  the  sum  and  difference  of  whose  geodesic  distances  from  C1 
and  Cg,  or  from  Ft  and  Fz,  are  constant.  In  the  latter  case  these 
curves  are  analogous  to  ellipses  and  hyperbolas  in  the  plane,  the 
points  Fl  and  F2  corresponding  to  the  foci.  For  this  reason  they 
are  called  geodesic  ellipses  and  hyperbolas,  which  names  are  given 
likewise  to  the  curves  u^  =  const.,  vl  =  const.,  when  the  distances 
are  measured  from  two  curves,  Cl  and  C2.  From  (60)  follows  at 
once  the  theorem  of  Weingarten :  * 

A  system  of  geodesic  ellipses  and  hyperbolas  is  orthogonal. 

By  means  of  (61)  equation  (60)  can  be  transformed  into  (59),  thus 
proving  that  when  the  linear  element  of  a  surface  is  in  the  form  (60), 
the  parametric  curves  are  geodesic  ellipses  and  hyperbolas. 

If  6  denotes  the  angle  which  the  tangent  to  the  curve  v^=  const, 
through  a  point  makes  with  the  curve  v  =  const.,  it  follows  from 

(III,  23)  that  0  ft)  ...    « , 

cos  u  =  cos—  i         sin  0  =  sin  —  • 

Hence  we  have  the  theorem : 

Given  any  two  systems  of  geodesic  parallels  upon  a  surface ;  the 
corresponding  geodesic  ellipses  and  hyperbolas  bisect  the  angles 
included  by  the  former. 

91.  Surfaces  of  Liouville.  Dini  f  inquired  whether  there  were 
any  surfaces  with  an  isothermal  system  of  geodesic  ellipses  and 
hyperbolas.  A  necessary  and  sufficient  condition  that  such  a  sur 
face  exist  is  that  the  coefficients  of  (60)  satisfy  a  condition  of  the 
form  (§41)  ^ 8in2»  =  r/i CQ8, | , 

where  U^  and  Vr  denote  functions  of  ul  and  i\  respectively.  In 
this  case  the  linear  element  may  be  written 

V '  i      *\ 

*Ueber  die  Oberfliichen  fur  welche  einer  der  beiden  Hauptkrummungshalbmesser 
eine  Function  des  anderen  ist,  Crelle,  Vol.  LXII  (1863),  pp.  160-173. 
t  Annali,  Ser.  2,  Vol.  Ill  (1869),  pp.  269-293. 


SURFACES  OF  LIOUVILLE  215 

By  the  change  of  parameters  defined  by 


1 

this  linear  element  is  transformed  into 


(63)  ds*  =  (  U2  +  r8)  (du*  +  dvt), 

where  U2  and  Vz  are  functions  of  u2  and  v2  respectively,  such  that 


Conversely,  if  the  linear  element  is  in  the  form  (63),  it  may  be 
changed  into  (62)  by  the  transformation  of  coordinates 


Surfaces  whose  linear  element  is  reducible  to  the  form  (63)  were 
first  studied  by  Liouville,  and  on  that  account  are  called  surfaces  of 
Liouville.*  To  this  class  belong  the  surfaces  of  revolution  and  the 
quadrics  (§§  96,  97).  We  may  state  the  above  results  in  the  form  : 

When  the  linear  element  of  a  surface  is  in  the  Liouville  form,  the 
parametric  curves  are  geodesic  ellipses  and  hyperbolas  ;  these  systems 
are  the  only  isothermal  orthogonal  families  of  geodesic  conies.^ 

92.  Integration  of  the  equation  of  geodesic  lines.  Having  thus 
discussed  the  various  properties  of  geodesic  lines,  and  having  seen 
the  advantage  of  knowing  their  equations  in  finite  form,  we  return 
to  the  consideration  of  their  differential  equation  and  derive  certain 
theorems  concerning  its  integration. 

Suppose,  in  the  first  place,  that  we  know  a  particular  first  inte 
gral  of  the  general  equation,  that  is,  a  family  of  geodesies  defined 
by  an  equation  of  the  form 

(64) 


From  (IV,  58)  it  follows  that  M  and  N  must  satisfy  the  equation 
_2_  /  ___  FN-GM  \      d_  I  FM-EN  \  = 

du  \^/EN'2  -  2  FMN  +  GM'2/      ^  \^EN'2-  2  FMN  +  GM2/ 


*  Journal  de  Mathematiques,  Vol.  XI  (1846),  p.  345. 

t  The  reader  is  referred  to  Darboux,  Vol.  II,  p.  208,  for  a  discussion  of  the  conditions 
under  which  a  surface  is  of  the  Liouville  type. 


216  GEODESICS 

In  consequence  of  this  equation  we  know  that  there  exists  a  func 
tion  <f>  denned  by 

.„_.   dc#>  EN-FM  d<f>  FN-GM 

(DO  )  —  =  —  ======================  ?        —  ==:  —  -===============================  • 

du      ^EN*-1FMN+GM*          to      V  EN*-2FMN+GM* 

Moreover,  we  find  that 

(66)  A^=l. 

From  (III,  31)  and  (65)  it  follows  that  the  curves  </>  =  const,  are 
the  orthogonal  trajectories  of  the  given  geodesies,  and  from  (66) 
it  is  seen  that  <£  measures  distance  along  the  geodesies  from  the 
curve  $  =  0.  Hence  we  have  the  theorem  of  Darboux  *  : 

When  a  one-parameter  family  of  geodesies  is  defined  by  a  differ 
ential  equation  of  the  first  order,  the  finite  equation  of  their  orthogonal 
trajectories  can  be  obtained  by  a  quadrature,  which  gives  the  geodesic 
parameter  at  the  same  time. 

Therefore,  when  the  general  first  integral  of  the  equation  of 
geodesies  is  known,  all  the  geodesic  parallels  can  be  found  by 
quadratures. 

We  consider  now  the  converse  problem  of  finding  the  geodesies 
when  the  geodesic  parallels  are  known.  Suppose  that  we  have  a 
solution  of  equation  (66)  involving  an  arbitrary  constant  a,  which 
is  not  additive.  If  this  equation  be  differentiated  with  respect  to 
a,  we  get 

(67) 


where  the  differential  parameter  is  formed  with  respect  to  the  linear 
element.  But  this  is  a  necessary  and  sufficient  condition  (§  37)  that 
the  curves  </>  =  const,  and  the  curves 

(68)  ^  =  const.=  a' 

da 

form  an  orthogonal  system.  Hence  the  curves  defined  by  (68) 
are  geodesies.  In  general,  this  equation  involves  two  arbitrary 
constants,  a  and  a',  which,  as  will  now  be  shown,  enter  in  such 
a  way  that  this  equation  gives  the  general  integral  of  the  differ 
ential  equation  of  geodesic  lines. 

*  Lemons,  Vol.  II,  p.  430;  cf.  also  Bianchi,  Vol.  I,  p.  202. 


EQUATIONS  OF  GEODESIC  LINES  217 

Suppose  that  a  appears  in  equation  (68),  and  write  the  latter  thus  : 

(69)  f  (u,  v,  a)  =  ar, 
in  which  case  equation  (67)  becomes 

(70)  Al(*,^)=0. 

The  direction  of  each  of  the  curves  (69)  is  given  by  -^-  /  —  £  •  If  this 

'          36  /dd> 
ratio  be  independent  of  a,  so  also  by  (70)  is  the  ratio  -^-1  —  • 

Write  the  latter  in  the  form  ' 


If  this  equation  and  (66)  be  solved  for  —  and  —  »  we  obtain  values 

cu          dv 

independent  of  a,  so  that  a  would  have  been  additive.    Hence  / 

involves  a,  and  so  also  does  iJL/£l_,  and  therefore  a  direction  at 

cu  /  dv 

a  point  (MO,  v0)  determines  the  value  of  a;  call  it  «0.  If  then  a'Q 
be  such  that  ^.  («„„„,  *,)  =  4, 

the  geodesic  -v/r  (%,  v,  a0)  =  ^  passes  through  the  point  (w0,  ?;0)  and 
has  the  given  direction  at  the  point.  Hence  all  the  geodesies  are 
defined  by  equation  (68),  and  we  have  the  theorem: 

Criven  a  solution  of  the  equation  A1<^  =  1,  involving  an  arbitrary 

r\    II 

constant  a,  in  such  a  way  that  —  involves  a;  the  equation 

da 


da 

for  all  values  of  a'  is  the  finite  equation  of  the  geodesies,  and  the 
arc  of  the  geodesies  is  measured  by  (/>.* 

By  means  of  this  result  we  establish  the  following  theorem  due 
to  Jacobi  : 

If  a  first  integral  of  the  differential  equation  of  geodesic  lines  be 
known,  the  finite  equation  can  be  found  by  one  quadrature. 

Such  an  integral  is  of  the  form 

dv 

-—  =  ^(u,  v,  a), 
du 

»Cf.  Darboux,  Vol.  II,  p.  429. 


218  GEODESICS 

where  a  is  an  arbitrary  constant.    As  this  equation  is  of  the  form 
(64),  the  function  c/>,  defined  by 

'(#  + 


=  P 


is  a  solution  of  equation  (66).    As  $  involves  a  in  the  manner 
specified   in  the  preceding  theorem,   the   finite   equation  of   the 

d(f>        , 
geodesies  is  —  =  a. 

93.  Geodesies  on  surfaces  of  Liouville.  The  surfaces  of  Liouville 
(§  91)  afford  an  excellent  application  of  the  theorem  of  Jacobi. 
We  take  the  linear  element  in  the  form  * 

(71)  ds2  =  (U-  V)  (U?du2  +  V?dv2), 

which  evidently  is  no  more  general  than  (63).    In  this  case  equa 
tion  (66)  becomes 


When  this  equation  is  written  in  the  form 


u*\du. 

one  sees  that  it  belongs  to  the  class  of  partial  differential  equa 
tions  admitting  an  integral  which  is  the  sum  of  functions  of  u 
and  v  alone,  f  In  order  to  obtain  this  integral,  we  put  each  side 
equal  to  a  constant  a  and  integrate.  This  gives 

(72)  </>  =  C l\ -\/U—a du  ±  f  F!  Va  —  Vdv. 

Hence  the  equation  of  geodesies  is 

(73) 

If  6  denotes  the  angle  which  a  geodesic  through  a  point  makes 
with  the  line  v  =  const,  through  the  point,  it  follows  from  (III,  24) 

and  (71)  that  y  dv 

tan  6  =  — -  —  • 

*  Cf.  Darboux,  Vol.  Ill,  p.  9.  t  Forsyth,  Differential  Equations  (1888),  p.  310. 


SURFACES  OF  LIOUVILLE  219 

If  the  value  of  dv/du  from  equation  (73)  be  substituted  in  this 

equation,   we   obtain  the    following    first   integral  of   the   Gauss 

equation  (56): 

(74)  ?7sin20  +  Fcos20  =  a. 

This  equation  is  due  to  Liouville.  * 

EXAMPLES 

1.  On  a  surface  of  constant  curvature  the  area  of  a  geodesic  triangle  is  pro 
portional  to  the  difference  between  the  sum  of  the  angles  of  the  triangle  and 
two  right  angles. 

2.  Show  that  for  a  developable  surface  the  first  integral  of  equation  (56)  can 
be  found  by  quadratures. 

3.  Given  any  curve  C  upon  a  surface  and  the  developable  surface  which  is  the 
envelope  of  the  tangent  planes  to  the  surface  along  C;  show  that  the  geodesic 
curvature  of  C  is  equal  to  the  curvature  of  the  plane  curve  into  which  C  is  trans 
formed  when  the  developable  is  developed  upon  a  plane. 

4.  When  the  plane  is  referred  to  a  system  of  confocal  ellipses  and  hyperbolas 
whose  foci  are  at  the  distance  2  c  apart,  the  linear  element  can  be  written 


5.  A  necessary  and  sufficient  condition  that  0  be  a  solution  of  Ai0  =  1  is  that 
ds2  —  d<p'2  be  a  perfect  square. 

6.  If  0  =  did  +  62 ,  where  6\  and  62  are  functions  of  u  and  v,  is  a  solution  of 
Ai0  =:  1,  the  curves  0i  =  const,  are  lines  of  length  zero,  and  the  curves  B\a  -j-  62  =  const, 
are  their  orthogonal  trajectories. 

7.  When  the  linear  element  of  a  spiral  surface  is  in  the  form  ds2  =  e2"  (du2  -\-  U"2do2), 
the  equation  Ai0  =  1  admits  the  solution  e?'Z7i,  where  U\  is  a  function  of  M,  which 
satisfies  an  equation  of  the  first  order  whose  integration  gives  thus  all  the  geodesies 
on  the  surface. 

8.  For  a  surface  with  the  linear  element 


where  V  and  V\  are  functions  of  v  alone,  the  equation  Ai0  =  1  admits  the  solution 
(f>  —  u\fsi  (v)  -f  ^2  (v),  the  determination  of  the  functions  \f>i  and  ^2  requiring  the  solu 
tion  of  a  differential  equation  of  the  first  order  and  quadratures. 

9.  If  0  denotes  a  solution  of  Ai0  =  1  involving  a  nonadditive  constant  a,   the 
linear  element  of  the  surface  can  be  written 


ca 

where  ®(0,  $}  indicates  the  mixed  differential  parameter  (III,  48). 

*i.c.,p.  348. 


220  GEODESICS 

94.  Lines  of  shortest  length.  Envelope  of  geodesies.  We  can  go 
a  step  farther  than  the  first  theorem  of  §  89  and  show  that  whether 
one  or  more  geodesies  pass  through  two  points  Ml  and  M2  on  a  sur 
face,  the  shortest  distance  on  the  surface  between  these  points,  if  it 
exists,  is  measured  along  one  of  these  geodesies. 

Thus,  let  v  =f(u)  and  v  =fl(u)  define  two  curves  C  and  Cl  passing 
through  the  points  M^  M#  the  parametric  values  of  u  at  the  points 
being  u^  and  u2.  The  arc  of  C  between  these  points  has  the  length 


(75)  •  =  £ 


2Fv'+Gv'2du, 


where  v'  denotes  the  derivative  of  v  with  respect  to  u.  For  con 
venience  we  write  the  above  thus : 

(76)  s=  f  *4>(ui  v,  v')du. 

Jiti 
Furthermore,  we  put 

f1(u)=f(u)  +  ea>(u), 

where  w(u)  is  a  function  of  u  vanishing  when  u  is  equal  to  ul  and 
M2,  and  e  is  a  constant  whose  absolute  value  may  be  taken  so  small 
that  the  curve  Cl  will  lie  in  any  prescribed  neighborhood  of  C. 
Hence  the  length  of  the  arc  M1M2  of  Cl  is 


= 

fc/tt 


(u,  v  -f-  e  tw,  v'  -f-  e  CD')  C?M. 


Thus  «j  is  a  function  of  e,  reducing  for  e  =  0  to  s.  Hence,  in  order 
that  the  curve  C  be  the  shortest  of  all  the  neai?-by  curves  which 
pass  through  Ml  and  J/2,  it  is  necessary  that  the  derivative  of  sl 
with  respect  to  e  be  zero  for  e  =  0.  This  gives 


On  the  assumption  that  «  admits  a  continuous  first  derivative 
in  the  interval  (u^  uz),  and  <f>  continuous  first  and  second  deriva 
tives,  the  left-hand  member  of  this  equation  may  be  integrated 
by  parts  with  the  result 


"1  /&£       d  d<l>\  ,        n 
wl-2- ^lauaaO: 

\v      du   v' 


LINES  OF  SHORTEST  LENGTH 


221 


for  o>  vanishes  when  u  equals  u^  and  u2.  As  the  function  &>  is  arbi 
trary  except  for  the  above  conditions  upon  it,  this  equation  is 
equivalent  to  the  following  equation  of  Euler  *  : 


(77) 


du 


When  this  result  is  applied  to  the  particular  form  of  (f>  in  equa- 
tkm  (75),  we  have 


d 


F+  Gv' 


__     I      •/   _  71'     I      _  ?? 

dv          cv          dv 


_  ~ 

" 


which  is  readily  reducible  to  equation  (43). 

Hence  the  shortest  distance  between  two  points,  if  existent,  is 
measured  along  a  geodesic  through  the  points.  This  geodesic  is 
unique  if  the  surface  has  negative  total  curvature  at  all  points. 
For  other  surfaces  more  than  one  geo 
desic  may  pass  through  the  points  if 
the  latter  are  sufficiently  far  apart.  We 
shall  now  investigate  the  nature  of  this 
problem. 

Let  v  —f(u,  a)  define  the  family  of  geo 
desies  through  a  point  J/0(w0,  v0),  and  let 
v  =  g  (u)  be  the  equation  of  their  envel 

ope  (o.  We  consider  two  of  the  geodesies  Cl  and  C2  (fig.  21),  and 
let  MI(UV  vj  and  M,,(u0,  vz)  denote  their  points  of  contact  with  the 
envelope.  Suppose  that  the  arc  M0M2  is  greater  than  Jf0Jfr  The 


distance  from  MQ  to 


^  measured  along  Cl  and  <£~is  equal  to 


D 


=  f 

JttQ 


f 

J^ 


If  3/2  is  considered  fixed  and  Ml  variable,  the  position  of  the  latter 
is  determined  by  a.    The  variation  of  D  with  M1  is  given  by 


j 

da  JM=MI 


*  Methodus  inveniendi  lineas  curvas  maximi  minimive  proprietate  gaudentes,  chap,  ii, 
§  21  (Lausanne,  1744)  ;  cf.  Bolza,  Lectures  on  the  Calculus  of  Variations,  p.  22  (Chicago, 
1994). 


222  GEODESICS 

B u t  f or  u  =  u.,  f  =  g  and  f  =  g'\  consequently  the  last  term  is  zero. 

f)f 
Integrating  the  first  member  hy  parts,  and  noting  that  -f-  is  zero  for 

u  =  u0  and  u  =  ux  (§  26),  we  have 


Since  C^  is  a  geodesic,  the  expression  in  parenthesis  is  zero,  and 
hence  D  does  not  vary  with  Mr  This  shows  that  the  envelope  of 
the  geodesies  through  a  point  bears  to  them  the  relation  which 
the  evolute  of  a  curve  does  to  a  family  of  normals  to  the  curve. 
Moreover,  the  curve  (§"is  not  a  geodesic,  for  at  each  point  of  it  there 
is  tangent  a  geodesic.  Hence  there  is  an  arc  connecting  M^  and  M2 
which  is  shorter  than  the  arc  of  &.  In  this  way,  by  taking  different 
points  Ml  on  &  we  obtain  any  number  of  arcs  connecting  MQ  and 
Mz  which  are  shorter  than  the  arc  of  C2,  each  consisting  of  an  arc 
of  a  geodesic  such  as  Cl  and  the  geodesic  distance  MVM^  It  is  then 
necessarily  true  that  the  shortest  distance  from  MQ  to  a  point  M  of 
C2  beyond  M2  is  not  measured  along  <72.  However,  when  M  lies 
within  the  arc  MQM2,  a  domain  can  be  chosen  about  (72  so  small 
that  the  arc  MQM  of  <72  is  shorter  than  the  arc  MQM  of  any  other 
curve  within  the  domain  and  passing  through  these  points.* 

Another  historical  problem  associated  with  this  problem  is  the  following :  t 
Given  an  arc  C0  joining  two  points  A,  B  on  a  surface ;  to  find  the  curve  of  shortest 
length  joining  A  and  B,  and  inclosing  with  Co  a  given  area. 

The  area  is  given  by  CClfdudv.    It  is  evident  that  two  functions  M  and  N  can 
be  found  in  an  infinity  of  ways  such  that  r< 

__  8N      dM 
~  du        dv 

By  the  application  of  Green's  theorem  we  have 


lldudv  =- 


// 

where  the  last  integral  is  curvilinear  and  is  taken  around  the  contour  of  the  area. 
Since  CQ  is  fixed,  our  problem  reduces  to  the  determination  of  a  curve  C  along 
which  the  integral  C'*Mdu  +  Ndv  is  constant,  and  whose  arc  AB,  that  is,  the 

«/ A 

*  For  a  more  complete  discussion  of  this  problem  the  reader  is  referred  to  Darboux, 
Vol.  Ill,  pp.  86-112;  Bolza,  chap.  v. 

tin  fact,  it  was  in  the  solution  of  this  problem  that  Minding  (Crelle,  Vol.  V  (18.30), 
p.  297)  discovered  the  function  to  which  Bonnet  (Journal  de  I'Ecole  Poly  technique, 
Vol.  XIX  (1848),  p.  44)  gave  the  name  geodesic  curvature.. 


ENVELOPE  OF  GEODESICS  223 

integral  C  V2?  -f  2  Fv'  +  Gv'^du,  is  a  minimum.  From  the  calculus  of  variations 
we  know  that,  so  far  as  the  differential  equations  of  the  solution  is  concerned,  this 
is  the  same  problem  as  finding  the  curve  C  along  which  the  integral 

fB VE  +  2  Fv'  +  Gv'*du  +  c(M  +  Nv')du 
JA 

is  a  minimum,  c  being  a  constant.    Euler's  equation  for  this  integral  is 

«  +  >^+^.!? 

d  /  F  4-  GV          \      cto cu dy_  _ 


-f  6rv'2/        V E  +  2  .FV  -f  Crt/2 

Comparing  this  result  with  the  formula  of  Bonnet  (IV,  56),  we  see  that  C  has  con 
stant  geodesic  curvature  1/c,  and  c  evidently  depends  upon  the  magnitude  of  the 
area  between  the  curves.  Hence  we  have  the  theorem  of  Minding  :* 

In  order  that  a  curve  C  joining  two  points  shall  be  the  shortest  which,  together  with 
a  given  curve  through  these  points,  incloses  a  portion  of  the  surface  with  a  given  area, 
it  is  necessary  that  the  geodesic  curvature  of  C  be  constant. 

GENERAL  EXAMPLES 

1.  When  the  parametric  curves  on  the  unit  sphere  satisfy  the  condition 


12  )'       a  I  12  ) '      0  (  12 


1   J        dv'-   2 


12  ) '  j  12  )' 

i  n  2  r 


they  represent  the  asymptotic  lines  on  a  surface  whose  total  curvature  is 


2.  When  the  equations  of  the  sphere  have  the  form  (III,  35),  the  parametric 
curves  are  asymptotic  and  the  equation  (22)  is  (1  +  wu)2—  —  =  -  20,  of  which  the 

CU  vV 

general  integral  is 

^2^(K)  +  ^(.)_ 

1  +  uv 
where  0  (u)  and  \f/  (v)  denote  arbitrary  functions. 

3.  The  sections  of  a  surface  by  all  the  planes  through  a  fixed  line  L  in  space, 
and  the  curves  of  contact  of  the  tangent  cones  to  the  surface  whose  vertices  are 
on  L,  form  a  conjugate  system. 

4.  Given  a  surface  of  translation  x  =  u,  y  =  v,  z  =f(u)  +  0(0).  Determine  the 
functions/  and  0  so  that  (Pl  +  P2)Z  =  const.,  where  Z  denotes  the  cosine  of  the 
angle  which  the  normal  makes  with  the  z-axis,  and  determine  the  lines  of  curva 
ture  on  the  surface. 

5.  Determine  the  relations  between  the  exponents  m<  and  nt-  in  the  equations 

x  =  UmiVni,        y  =  Um*V"*t        z  =  Um3VHs, 

so  that  on  the  surface  so  defined  the  parametric  curves  shall  form  a  conjugate  sys 
tem,  and  show  that  the  asymptotic  lines  can  be  found  by  quadratures. 

*Z.c.,  p.  207. 


224  GEODESICS 

6.  The  envelope  of  the  family  of  planes 

(Ui  +  Fi)z  +  (Uz  +  V2)y  +  (Us  +  F8)z  +  (U*  +  F4)  =  0, 

where  the  U"'s  are  functions  of  u  alone  and  the  F's  of  »,  is  a  surface  upon  which 
the  parametric  curves  are  plane,  and  form  a  conjugate  system. 

7.  The  condition  that  the  parametric  curves  form  a  conjugate  system  on  the 
envelope  of  the  plane 

x  cos  u  +  y  sin  u  -f  z  cot  v  =/(u,  u), 

is  that  /  be  the  sum  of  a  function  of  u  alone  and  of  v  alone ;  in  this  case  these 
curves  are  plane  lines  of  curvature. 

8.  Find  the  geodesies  on  the  surface  of  Ex.  7,  p.  219,  and  determine  the  expres 
sions  for  the  radii  of  curvature  and  torsion  of  a  geodesic. 

9.  A  representation  of  two  surfaces  upon  one  another  is  said  to  be  conformal- 
conjugate  when  it  is  at  the  same  time  conformal,  and  every  conjugate  system  on 
one  surface  corresponds  to  a  conjugate  system  on  the  other.    Show  that  the  lines  of 
curvature  correspond  and  that  the  characteristic  lines  also  correspond. 

10.  Given  a  surface  of  revolution  z  =  ucosu,  y  =  wsinw,  z—f(u),  and  the 
function  0  defined  by 

(i) 

where  A  and  c  are  constants ;  a  conf orjual-conjugate  representation  of  the  surface 
upon  a  second  surface  x\  =  MI  cos«i,  y\  —  MI  sin  I?!,  z\  =  <f>(ui)  is  defined  by 


-         du 

V  —  CUi,  C  log  Ui  = 


where  F'  denotes  the  function  of  M  found  by  solving  (i)  for  <£'. 

11.  If   two   families  of   geodesies  cut  under  constant  angle,  the  surface  is 
developable. 

12.  If  a  surface  with  the  linear  element 

ds*  =  (aM2  -  bv2  -  c)  (du?  +  cto2), 

where  a,  6,  c  are  constants,  is  represented  on  the  xy-plane  by  u  =  x,  v  =  y,  the 
geodesies  correspond  to  the  Lissajous  figures  defined  by 


where  A,  -Z?,  C  are  constants. 

13.  When  there  is  upon  a  surface  more  than  one  family  of  geodesies  which, 
together  with  their  orthogonal  trajectories,  form  an  isothermal  system,  the  curva 
ture  of  the  surface  is  constant. 

14.  If  the  principal  normals  of  a  curve  meet  a  fixed  straight  line,  the  curve  is  a 
geodesic  on  a  surface  of  revolution  whose  axis  is  this  line.    Examine  the  case  where 
the  principal  normals  meet  the  line  under  constant  angle. 


GENERAL  EXAMPLES  225 

15.  A  representation  of  two  surfaces  upon  one  another  is  said  to  be  a  geodesic 
representation  when  to  a  geodesic  on  one  surface  there  corresponds  a  geodesic  on 
the  other.  Show  that  the  representation  is  geodesic  when  points  with  the  same 
parametric  values  correspond  on  surfaces  with  the  linear  elements 


where  the  IPs  are  functions  of  u  alone,  the  F's  of  v  alone,  and  h  is  a  constant. 

16.  A  surface  with  the  linear  element 

ds2  =  (w*  -  v4)  [0  /- 
where  0  is  any  function  whatever,  admits  of  a  geodesic  representation  upon  itself. 

17.  A  necessary  and  sufficient  condition  that  an  orthogonal  system  upon  a  sur 
face  may  be  regarded  as  geodesic  ellipses  and  hyperbolas  in  two  ways,  is  that  when 
the  curves  are  parametric  the  linear  element  be  of  the  Liouville  form  ;  in  this  case 
these  curves  may  be  so  regarded  in  an  infinity  of  ways. 

18.  Of  all  the  curves  of  equal  length  joining  two  points,  the  one  which,  together 
with  a  fixed  curve  through  the  points,  incloses  the  area  of  greatest  extent,  has  con 
stant  geodesic  curvature. 

19.  Let  T  be  any  curve  upon  a  surface,  and  at  two  near-by  points  P,  P'  draw 
the  geodesies  g,  g'  perpendicular  to  T;  let  C  be  the  curve  through  P  conjugate 
to  gr,  P"  the  point  where  it  meets  g',  and  Q  the  intersection  of  the  tangents  to  g 
and  g'  at  P  and  P"  ;  the  limiting  position  of  Q,  as  Pf  approaches  P,  is  the  center 
of  geodesic  curvature  of  T  at  P. 

20.  Show  that  if  a  surface  S  admits  of  geodesic  representation  upon  a  plane  in 
such  a  way  that  four  families  of  geodesies  are  represented  by  four  families  of  par 
allel  lines,  each  geodesic  on  the  surface  is  represented  by  a  straight  line  (cf  .  Ex.  3, 
p.  209). 


CHAPTER  VII 

QUADRICS.     RULED  SURFACES.     MINIMAL  SURFACES 

95.  Confocal  quadrics.  Elliptic  coordinates.  Two  quadrics  are 
confocal  when  the  foci,  real  or  imaginary,  of  their  principal  sec 
tions  coincide.  Hence  a  family  of  confocal  quadrics  is  defined  by 
the  equation 

a)  -A+/-+-A-1, 

a2—  u      b2—u      c2—u 

where  u  is  the  parameter  of  the  family  and  a,  6,  c  are  constants, 
such  that 

(2)  a2  >  b2  >  c2. 

For  each  value  of  u,  positive  or  negative,  less  than  a2,  equation 
(1)  defines  a  quadric  which  is 

Ian  ellipsoid  when  c2  >  u  >  —  oo, 
an  hyperboloid  of  one  sheet  when  b2  >  u  >  c*, 
an  hyperboloid  of  two  sheets  when  a2  >  u  >  b2. 

As  u  approaches  c2  the  smallest  axis  of  the  ellipsoid  approaches 
zero.  Hence  the  surface  u  =  c2  is  the  portion  of  the  zy-plane, 
counted  twice,  bounded  by  the  ellipse 

(4)  2 


a2—  c2      b2—c2 


Again,  the  surface  u  =  b2  is  the  portion  of  the  ^-plane,  counted 
twice,  bounded  by  the  hyperbola 


which  contains  the  center  of  the  curve.    Equations  (4)  and  (5) 
define  the  focal  ellipse  and  focal  hyperbola  of  the  system. 


CONFOCAL  QUADEICS 


227 


Through  each  point  (x,  y,  z)  in  space  there  pass  three  quadries 
of  the  family;  they  are  determined  by  the  values  of  w,  which  are 
roots  of  the  equation 

(6)  <£  (u)  =  (a2  -  u)  (b2  -  u)  (c2  -  u)  -  x2  (b2  -  u)  (c2  -  u) 

-  y2(a2-u)  (c2-  u)  -  z2(a2-  u)  (b2-  u)  =  0. 

Since        </>  (a2)  <  0,     c/>  (b2)  >  0,     <£  (c2)  <  0,     </>  (-  oo)  >  0, 

the  roots  of  equation  (6),  denoted  by  ul9  u0,  w3,  are  contained  in 
the  following  intervals : 


(7) 


a2  >  u,  >  b2,      b2  >u,>  c2,      c2  >  u^>  — 


oo. 


From  (3)  it  is  seen  that  the  surfaces  corresponding  to  uv  w2,  us  are 
respectively  hyperboloids  of  two  and  one  sheets  and  an  ellipsoid. 

Fig.  22  represents  three  confocal  quadrics;   the  curves  on  the 
ellipsoid   are   lines    of    cur 
vature,  and   on   the   hyper- 
boloid  of  one  sheet  they  are 
asymptotic  lines. 

From  the  definition  of  uv 
w2,  us  it  follows  that  <£  (u)  is 
equal  to  (u^—u)  (u2—u)  (us—u). 
When  (/>  in  (6)  is  replaced 
by  this  expression  and  u 
is  given  successively  the 
values  a2,  b2,  c2,  we  obtain  * 


FIG.  22 


(8) 


or  = 


= 


(*-«')(€>-?) 


These  formulas  express  the  Cartesian  coordinates  of  a  point  in 
space  in  terms  of  the  parameters  of  the  three  quadrics  which 
pass  through  the  point.  These  parameters  are  called  the  elliptic 
coordinates  of  the  point.  It  is  evident  that  to  each  set  of  these 


*  Kirchhoff ,  Mechanik,  p.  203.    Leipsic,  1877. 


228 


QUABEICS 


coordinates  there  correspond  eight  points  in  space,  one  in  each 
of  the  eight  compartments  bounded  by  the  coordinate  planes. 

If  one  of  the  parameters  «-t.  in  (8)  be  made  constant,  and  the 
others  u^  %,  where  i  =£  j  =£  Ar,  be  allowed  to  vary,  these  equations 
define  in  parametric  form  the  surface,  also  defined  by  equation 
(1),  in  which  u  has  this  constant  value  ur  The  parametric  curves 
1^.=  const.,  uk—  const,  are  the  curves  of  intersection  of  the  given 
quadric  and  the  double  system  of  quadrics  corresponding  to  the 
parameters  Uj  and  uk, 

If  we  put 

,r\\  o  12  7*  2  «.  ng     .. 

the  equation  of  the  surface  becomes 

(10)  -  +  ^  +  -=1, 

a       b       c 

and  the  parametric  equations  (8)  reduce  to 


(11) 


la(a  —  u)(a  —  v) 
"'  N  (a  -  b)  (a  -  c)  ' 

fb  (b  —  u)(b  —  v) 
y~~~  \  (b  -  a)  (b  -  c)  ' 

\c(c  —  u)  (c  —  v) 
Z~~  N  (c—  a}(c  —  b} 


Moreover,  the  quadrics  which  cut  (10)  in  the  parametric  curves 
have  the  equations: 


(12) 


a  —  u      b  —  u      c  —  u 


a—v      b—v      c—v 


=  1, 


=  1. 


In  consequence   of   (3)   and  (9)   we  have  that  equations   (10) 
or  (11)  define 

an  ellipsoid  when  a>u>b>v>c>0, 
(13)  -  an  hyperboloid  of  one  sheet  when  a>u>b>Q>c>v, 

an  hyperboloid  of  two  sheets  when  a>0>b>u>c>v. 


FUNDAMENTAL  QUANTITIES 


229 


96.  Fundamental  quantities  for  central  quadrics.    By  direct  cal 
culation  we  find  from  (11) 

*.        U(U-V)  ™_A  ^_V(V-U) 


(14) 


/(¥) 


where  for  the  sake  of  brevity  we  have  put 
(15)  f(0)  =  4  (a  -  6)  (b  -  0)  (c  -  6). 

We  derive  also  the  following  : 


(16) 


and 

(17) 


(a-b)(a- 


=      \ 

^ 


—  a)(c  —  b) 


lobe  u  —  v 
JL>  —  —  \  


abc  u  —  v 


uv  f(u)  N  uv  f(v) 

Since  F  and  D'  are  zero,  the  parametric  curves  are  lines  of  curva 
ture.  And  since  the  change  of  parameters  (9)  did  not  change  the 
parametric  curves,  we  have  the  theorem : 

The  quadrics  of  a  confocal  system  cut  one  another  along  lines  of 
curvature,  and  the  three  surfaces  through  a  point  cut  one  another 
orthogonally  at  the  point. 

This  result  is  illustrated  by  fig.  22. 
From  (14)  and  (17)  we  have 

.,£  1  lobe  1  _  \abc  1     _  abc 

Pl~       NtfV'        p2~       N^3'        />^2~wV' 

Hence  the  ellipsoid  and  hyperboloid  of  two  sheets  have  positive 
curvature  at  all  points,  whereas  the  curvature  is  negative  at  all 
points  of  the  hyperboloid  of  one  sheet. 
If  formulas  (16)  be  written 


\abc  x 
uv  a 


abc  z 
uv  c 


>  uv  a  \  uv  b 

the  distance  W  from  the  center  to  the  tangent  plane  is 


(19) 


230  QUADEICS 

Hence : 

The  tangent  planes  to  a  central  quadric  along  a  curve,  at  points 
of  which  the  total  curvature  of  the  surface  is  the  same,  are  equally 
distant  from  the  center. 

From  (18)  we  see  that  the  umbilical  points  correspond  to  the 
values  of  the  parameters  such  that  u  =  v.  The  conditions  (13) 
show  that  this  common  value  of  u  and  v  for  an  ellipsoid  is  b, 
and  c  for  an  hyperboloid  of  two  sheets,  whereas  there  are  no  real 
umbilical  points  for  the  hyperboloid  of  one  sheet.  When  these 
values  are  substituted  in  (11),  we  have  as  the  coordinates  of 
these  points  on  the  ellipsoid 

\c(b-c) 


and  on  the  hyperboloid  of  two  sheets 


It  should  be  noticed  that  these  points  lie  on  the  focal  hyperbola 
and  focal  ellipse  respectively. 

97.  Fundamental  quantities  for  the  paraboloids.    The  equation 
of  a  paraboloid 

(22)  2z  =  ax2+by* 
may  be  replaced  by 

(23)  a:=V^,         y=V^,         z  =  -(aul+bvl). 

Hence  the  paraboloids  are  surfaces  of  translation  (§  81)  whose 
generating  curves  are  parabolas  which  lie  in  perpendicular  planes. 
By  direct  calculation  we  find 


D'  =  0,      //'=-- 

^VS^  +  ftX  +  l  4^ 

so  that  the  equation  of  the  lines  of  curvature  is 

a  dv,      b      dv.   .  b 


FUNDAMENTAL  QUANTITIES 
The  general  integral  of  this  equation  is 
(24)  ^ 


231 


where  c  is  an  arbitrary  constant. 

When  ul  and  v1  in  (24)  are  given  particular  values,  equation  (24) 
determines  two  values  of  c,  cl  and  £2,  in  general  distinct.  If  these 
latter  values  be  substituted  in  (24)  successively,  we  obtain  in  finite 
form  the  equations  of  the  two  lines  of  curvature  through  the  point 

/  \        Tf  ,1  "U  1/11  A  +  au\  J  A  +  aV 

(».,  tu.    11  cl  and  cz  be  replaced  by  —  (  —  -  -  )  and  —  (  —  — 

\     on    I  \    ov 

spectively,  we  have,  in  consequence  of  (23),  the  two  equations 


re- 


(25) 


buy2  +  (1  +  au)  x2  =  u  (1  +  an} 

ab 


-f  (1  -f-  av)  x2  =  v  (1  +  av) 


b-a 
ab 


When  these  equations  are  solved  for  x2  and  y1,  we  find  that  equa 
tion  (22)  can  be  replaced  by 

a  —  b 
b 

(26) 


1  b  ~  a  /i  , 

/  =  2"^-(1+aW  +  ^' 

and  the  parametric  curves  are  the  lines  of  curvature. 
Now  we  have 


(27) 


a  —  b  a(a  —  b)u  — 

E  —  ——r—  (u  —  v)  —  *—  —  ,       F  =  0, 

2 


b  —  a 

G  =  ~7T5~  (U  ~ 

4  62 


u(I+au) 
a(a  —  b)v  — 


au}(\  +  av),  —Vab 


and 


(29) 


V  [a  (a  —  b)  u  —  b]  [a  (a  —  b)  v  —  b] 

(a-b)(u-v)  1 


1     U3 


[a(a-b)u-b][a(a-b)v-b] 


(a  —  b)(u  —  v) 


a(a  —  b)u  —  b][a(a  —  b)v-  b]    v(l+av) 


232  QLJADRICS 

From  (27),  (28),  and  (29)  we  obtain 

(30)  W  =  ^Xx  =  - 

[a  (a  -  b)u-  b]*[a(a  -  b)v- 
and 


(31) 


From  these  results  we  find  that  the  ratio  W/z  is  constant  along 
the  curves  for  which  the  total  curvature  is  constant. 

We  suppose  that  b  is  positive  and  greater  than  a.  From  the 
first  of  (26)  it  follows  that  u  and  v  at  a  real  point  differ  in 
sign,  or  one  is  equal  to  zero.  We  consider  the  points  at  which 
both  u  and  v  are  equal  to  zero.  There  are  two  such  points, 
and  their  coordinates  are 

(32)  ,-0, 


=  [a(a  —  b)u  —  b]~  [a(a-b)v- 


Evidently  these  points  are  real  only  on  the  elliptic  paraboloid. 
From  (31)  it  follows  that  pl  and  pz  are  then  equal,  and  conse 
quently  these  are  the  umbilical  points.  Since  at  points  other 
than  these  u  and  v  must  differ  in  sign,  we  may  assume  that  u 
is  always  positive  and  v  negative.  Moreover,  from  (26)  it  is 
seen  that  u  and  v  are  unrestricted  except  in  the  case  of  the 
elliptic  paraboloid,  when  v  must  be  greater  than  —  I/a. 

98.  Lines  of  curvature  and  asymptotic  lines  on  quadrics.  From 
(14),  (27),  and  §  91  we  have  the  theorem  : 

The  lines  of  curvature  of  a  quadric  surface  form  an  isothermal 
system  of  the  Liouville  type. 

Bonnet  *  has  shown  that  this  property  is  characteristic  of  the 
quadrics.  There  are,  however,  many  surfaces  whose  lines  of  curva 
ture  form  an  isothermal  system.  They  are  called  isothermic  sur 
faces.  The  complete  determination  of  all  such  surfaces  has  never 
been  accomplished  (cf.  Ex.  3,  §  65). 

*  Meraoire  sur  la  theorie  des  surfaces  applicables  sur  une  surface  donne'e,  Journal  de 
V  Ecole  Poly  technique,  Vol.  XXV  (1867),  pp.  121-132. 


ASYMPTOTIC  LINES  ON  QUADRICS  233 

From  (17),  (29),  and  §  82  follows  the  theorem: 

The  lines  of  curvature  of  a  quadric  surface  form  an  isothermal- 
conjugate  system,  and  consequently  the  asymptotic  lines  can  be  found 
by  quadratures. 

We  shall  find  the  expressions  for  the  coordinates  in  terms  of 
the  latter  in  another  way. 

Equation  (10)  is  equivalent  to  the  pair  of  equations 

\vS         V<y         \        Vo/       \V#        V< 
or  the  pair 

(34) 


where  u  and  v  are  undetermined.    For  each  value  of  u  equations 

(33)  define  a  line  all  of  whose  points  lie  on  the  surface.    And  to 
each  point  on  the  surface  there  corresponds  a  value  of  u  determin 
ing  a  line  through  the  point.    Hence  the  surface  is  ruled,  and  it  is 
nondevelopable,   as  seen  from  (18).    Again,  for  each  value  of  v 
equations  (34)  define  a  line  whose  points  lie  on  the  surface  (10), 
and  these  lines   are   different  from  those   of   the   other  system. 
Hence  the  central  quadrics   are  doubly  ruled.     These   lines   are 
necessarily  the  asymptotic  lines.    Consequently,  if  equations  (33), 

(34)  be  solved  for  z,  y,  z,  thus : 

x        u  +  v  y       uv  —  1  z        .  v  —  u 

V^i^r+i'      vP^TT'      vP'^TT' 

we  have  the  surface  defined  in  terms  of  parameters  referring  to 
the  asymptotic  lines. 

In  like  manner  equation  (22)  may  be  replaced  by 

^fax  -f  i^/by  =  2  uz,          ^/ax  —  i ^Jby  =  - , 
or  u 

V 'ax  +  i^Tby  —  -  •>  V 'ax  —  i  Vfo/  =±=  2  vz. 

v 

Solving  these,  we  have 

I  *  /  \  "1 

\          *  C%  tJ  c\  """" 


2uv 


234  QUADKICS 


As  in  the  preceding  case,  we  see  that  the  surface  is  doubly  ruled,* 
and  the  parameters  in  (36)  refer  to  the  asymptotic  system  of  straight 
lines.  Hence  : 

The  asymptotic  lines  on  any  quadric  are  straight  lines. 


EXAMPLES 

1.  The  focal  conies  of  a  family  of  confocal  quadrics  meet  the  latter  in  the 
umbilical  points. 

2.  Find  the  characteristic  lines  on  the  quadrics  of  positive  curvature. 

3.  The  normal  section  of  an  ellipsoid  at  a  point  in  the  direction  of  the  curve 
along  which  the  total  curvature  is  constant  is  an  ellipse  with  one  of  its  vertices 
at  the  point. 

4.  Find  the  equation  of  the  form  —  —  =  Md  (cf .  §  79)  when  the  corresponding 

cu  dv 

surface  is  a  hyperboloid  of  one  sheet ;  when  a  hyperbolic  paraboloid. 

5.  Find  the  evolute  of  the  hyperboloid  of  one  sheet  and  derive  the  following 
properties : 

(a)  the  surface  is  algebraic  of  the  twelfth  order ; 

(b)  the  section  by  a  principal  plane  of  the  hyperboloid  consists  of  a  conic  and 
the  evolute  of  a  conic  ; 

(c)  these  sections  are  edges  on  the  surface ; 

(d)  the  curve  of  intersection  of  the  two  sheets  of  the  surface  is  cut  by  each  of 
the  principal  planes  in  four  ordinary  points,  four  double  points,  and  four  cusps, 
and  consequently  is  of  the  twenty-fourth  order. 

6.  Determine  for  the  evolute  of  a  hyperbolic  paraboloid  the  properties  analogous 
to  those  for  the  surface  of  Ex.  5. 

7.  Deduce  the  equations  of  the  surfaces  parallel  to  a  central  quadric ;  determine 
their  order  and  the  character  of  the  sections  of  the  surface  by  the  principal  planes 
of  the  quadric  ;  find  the  normal  curvature  of  the  curves  corresponding  to  the  asymp 
totic  lines  on  the  quadric. 

99.  Geodesies  on  quadrics.  Since  the  quadrics  are  isothermic 
surfaces  of  the  Liouville  type,  the  finite  equation  of  the  geodesies 
can  be  found  by  quadratures  (§  93).  From  (VI,  74),  (14)  and  (27), 

*  Moreover,  the  quadrics  are  the  only  doubly  ruled  surfaces.  For  consider  such  a  sur 
face,  and  denote  by  a,  b,  c  three  of  the  generators  in  one  system.  A  plane  a  through  a 
meets  6  and  c  in  unique  points  B  and  C',  and  the  line  B(J  meets  a  in  a  point  A.  The  line 
ABC  is  a  generator  of  the  second  system,  and  the  only  one  of  this  system  in  the  plane  a. 
The  other  lines  of  this  system  meet  a  in  the  line  a.  On  this  account  the  plane  a  cuts  the 
surface  in  two  lines,  a  and  ABC,  that  is,  in  a  degenerate  conic.  Hence  the  surface  is  of 
the  second  degree. 


GEODESICS  ON  QUADEICS  235 

it  follows  that  the  first  integral  of  the  differential  equation  of 
geodesies  on  any  one  of  the  quadrics  is 

(37)  u  sin20  +  v  cos20  =  a, 

where  a  is  a  constant  of  integration  and  6  measures  the  angle 
which  a  geodesic,  determined  by  a  value  of  a,  makes  with  the  lines 
of  curvature  v  =  const.  We  recall  that  in  equations  (11)  and  (26) 
the  parameter  u  is  greater  than  v,  except  at  the  umbilical  points, 
where  they  are  equal.  We  shall  discuss  the  general  case  first. 

Consider  a  particular  point  M ' (u\  v').  According  as  a  is  given 
the  value  u1  or  vf,  equation  (37)  defines  the  geodesic  tangent  at 
M1  to  the  line  of  curvature  u  =  u'  or  v  —  v'  respectively.  It  is 
readily  seen  that  the  other  values  of  a,  determining  other  geo 
desies  through  M.\  lie  in  the  interval  between  u'  and  v'.  More 
over,  to  each  value  of  a  in  this  domain  there  correspond  two 
geodesies  through  Mr  whose  tangents  are  symmetrically  placed 
with  respect  to  the  directions  of  the  lines  of  curvature.  From 
this  result  it  follows  also  that  the  whole  system  of  geodesies  is 
defined  by  (37),  when  a  is  given  the  limiting  values  of  u  and  v 
and  all  the  intermediate  values. 

We  write  equation  (37)  in  the  form 

(38)  (u  —  a)  sin20  +  (v—a)  cos20  =  0, 

and  consider  the  geodesies  on  a  central  quadric  defined  by  this 
equation  when  a  has  a  particular  value  a'.  Suppose,  first,  that  a' 
is  in  the  domain  of  the  values  of  u.  Then  at  each  point  of  these 
geodesies  v<  a'  and  consequently  from  (38)  u  >  a'.  We  have  seen 
that  these  geodesies  are  tangent  to  the  line  of  curvature  u  =  a'- 
From  (11)  it  follows  that  they  lie  within  the  zone  of  the  surface 
bounded  by  the  two  branches  of  the  curve  u  =  a'.  When,  now, 
a'  is  in  the  domain  of  the  values  of  v,  u  —  a1  is  positive,  and  con 
sequently  from  (38)  v  <  a'.  Hence  the  geodesies  tangent  to  the 
curve  v  —  a'  lie  outside  the  zone  bounded  by  the  two  branches  of 
the  line  of  curvature  v  —  af.  Similar  results  are  true  for  the  parabo 
loids,  with  the  difference,  as  seen  from  (26),  that  the  geodesies 
tangent  to  u  —  a'  lie  outside  the  region  bounded  by  this  curve, 
whereas  the  curves  tangent  to  v  =  a'  lie  inside  the  region  bounded 
by  v  =  a'. 


236  QUADKICS 

100.  Geodesies  through  the  umbilical  points.  There  remains  for 
consideration  the  case  where  a  takes  the  unique  value  which  u 
and  v  have  at  the  umbilical  points.  Let  it  be  denoted  by  «0,  so 
that  the  curves  defined  by 

(39)  (u  —  aQ)  sin20  +  (v  —  a0)  cos20  =  0 

are  the  umbilical  geodesies.     We  have,  at  once,  the  theorem : 

Through  each  point  on  a  quadric  with  real  umbilical  points  there 
pass  two  umbilical  geodesies  which  are  equally  inclined  to  the  lines 
of  curvature  through  the  point. 

Hence  two  diametrically  opposite  umbilical  points  of  an  ellipsoid 
are  joined  by  an  infinity  of  geodesies,  and  no  two  geodesies  through 
the  same  umbilical  point  meet  again  except  at  the  diametrically 
opposite  point.  These  properties  are  possessed  also  by  a  family  of 
great  circles  on  a  sphere  through  two  opposite  points.  On  the 
elliptic  paraboloid  and  on  each  sheet  of  the  hyperboloid  of  two 
sheets  there  are  two  families  of  umbilical  geodesies,  but  no  two 
of  the  same  family  meet  except  at  the  umbilical  point  common  to 
all  curves  of  the  family. 

For  the  ellipsoid  (11)  «0=  b  and  equations  (VI,  72,  73)  become 


a^_  _1    C   \  u       ~~    du        1    C   \  v  _dv 

~db~  ~4J  ^\(a  —  u)(u—c)u  —  b     4  J  N(a  —  v)(v  —  c)  v  — 


Similar  results  hold  for  the  hyperboloid  of  tw£  sheets  and  the 
elliptic  paraboloid.  Hence  the  distances  of  a  point  P  from  two 
umbilical  points  (not  diametrically  opposite)  are  of  the  form 


Hence  we  have  : 

The  lines  of  curvature  on  the  quadrics  with  real  umbilical  points 
are  geodesic  ellipses  and  hyperbolas  with  the  umbilical  points  for  foci. 

101.  Ellipsoid  referred  to  a  polar  geodesic  system.  A  family  of 
umbilical  geodesies  and  their  orthogonal  trajectories  constitute 
an  excellent  system  for  polar  geodesic  coordinates,  because  the 
domain  is  unrestricted  (§87)  except  in  the  case  of  the  ellipsoid, 


UMBILICAL  GEODESICS 


237 


and  then  only  the  diametrically  opposite  point  must  be  excluded. 
We  consider  such  a  system  on  the  ellipsoid,  and  let  0  denote  the 
pole  of  the  system  and  0',  O",  0'"  the  other  umbilical  points  (fig.  23). 
If  we  put 


(40) 


i  r  I       «        ,     i  r  r     v 

<h  =  -  IA du  —  -  l\ 

2J   \(a-u)(u-c)  2J  N(a  — *)(»  — 


_  1    r  I  u  du     _  1    C   I  v 

~2  J  \(a  —  u)(u  —  c)  u  —  b~2jv(a—v)(v  — 

it  is  readily  found  that 


00 

1 


dv 

C)V  —  ( 


(j_. „)(&_„) 

By  means  of  (11)  we  may  reduce  the  linear  element  to  the  form 
(41)  ds2  =  dp  +  — '-= 


In  order  that  the  coordinates  be  polar  geodesic,  ^r  must  be 
replaced  by  another  parameter  measuring  the  angles  between 
the  geodesies.  For  the  ellipsoid 
equation  (39)  is 

(42)  (u-b)s 


FIG.  23 


As  previously  seen,  6  is  half  of 

one  of  the  angles  between  the 

two  geodesies  through  a  point 

M.    As  M  approaches  0  along 

the  geodesic  joining  these  two 

points,  the  geodesic  O'MO"'  ap 

proaches  the  section  #  =  0.   Consequently  the  angle  2  0  approaches 

the  angle  MOO1,  denoted  by  &>,  or  its  supplementary  angle.    Hence 

we  have  from  (42) 

(43) 


ib-V\ 
im = 

b,r=b  \U—  b/ 


lim 

u=b,  ?•  = 


We  take  &>  in  place  of  t/r  and  indicate  the  relation  between  them 
by  ^fr  =/(o>).    From  (41)  we  have 


238  QUADRICS 

This   expression  satisfies   the  first  of  conditions  (VI,  54).     The 
second  is 

(44)         lim  1  -7=^—  [<*  -  *)  §  -  («  -  •>  £1  -  1 

-j  2  7^-6)^-1.)  [  ^  *<H 

If  we  make  use  of  the  formulas  (III,  11)  and  (40),  we  find 


du       n     \(a  —  u)(u  —  c)u  —  b       dv  0     \(a  —  v)(v  —  c)  b  —  v 

-  =  %   -yj-  --       —  i      -    —  =  —  £t   \\  --       —  j 

d<p  M  ^  u  —  vd(f)  M  v  w  —  v 

so  th.it  equation  (44)  reduces  to 

lim  m  V(^M^)  r  |(c»-»o(u-g)  +  i(a-^(.-^i  =1 

u=»,r  =  b  U-V  \_\  U  N  V  J 

By  means  of  (43)  we  pass  from  this  to 


Hence  the  linear  element  has  the  following  form  due  to  Roberts  * : 

siir&> 
The  second  of  equations  (40)  may  now  be  put  in  the  form 

1  fjl     HI  ««        1   fjl     HI  dv 

2J  \l(ar.u)(p  —  c)  u  —  b      2J   \(a  —  v)(v- 

b  &)       ~ 

log  tan  -  -f  (7, 


I  (a  -  b)  (b  -  c) 

\vhere  C  denotes  the  constant  of  integration.  In  order  to  evaluate 
this  constant,  we  consider  the  geodesic  through  the  point  (0,  ft,  0). 
At  this  point  the  parameters  have  the  values  u  =  a,  v  =  <?,  and  the 
angle  co  has  a  definite  value  o>.  Hence  the  above  equation  may  be 
replaced  by 

i  r  r  ~^~     du  _i  r  r  ~^~     ^ 


(«-*)<*-«)   ,    2 

.    *  Journal  de  Mathematiques,  Vol.  XIII  (1848),  pp.  1-11. 


PROPERTIES  OF  QUADRICS  239 

In  like  manner,  for  the  umbilical  geodesies  through  one  of  the 
other  points  (not  diametrically  opposite)  we  have 


i  r  I      u         du  [  i  r  \       v        dv 

2Ja    \(a  —  u)(u  —  c)  u  —  b      2jc    \(a  —  v)(v  —  c)  v  — 


(a-b)(b-c) 


It  follows  at  once  from  these  formulas  that  if  M  is  any  point  on 
a  line  of  curvature  u  —  const,  or  v  —  const.,  we  have  respectively 

tan  —  -  --  tan  —  -  —  =  const.,        tan  —  -  —  cot  —  -  —  =  const. 

102.  Properties  of  quadrics.    From  (18)  it  follows  that  for  the 
central  quadrics  Euler's  equation  (IV,  34)  takes  the  form 


By  means  of  (19)  and  (37)  this  reduces  to 

(47)  I—?' 

R  abc 

In  like  manner,  we  have  for  the  paraboloids 

(48)  I  =  _J!  [&  +  „«(&_„)]. 

Hence  we  have  : 

Along  a  geodesic  or  line  of  curvature  on  a  central  quadric  the 
product  RW*  is  constant,  and  on  a  paraboloid  the  ratio  fiW3/zs. 

Consider  any  point  P  on  a  central  quadric  and  a  direction 
through  P.  Let  a,  ft,  7  be  the  direction-cosines  of  the  latter. 
The  semi-diameter  of  the  ellipsoid  (10)  parallel  to  this  direction  is 
given  by 

(49)  =  a-  +  . 


By  definition 


240  QUADRICS 

and  similarly  for  /3  and  7.     When  the  values  of  #,  «/,  2,  E,  G  from 
(11)  and  (14)  are  substituted,  equation  (49)  reduces  to 

1  =  cos2fl      sin2fl 

p*        u  v 

By  means  of  (19)  and  (37)  this  may  be  reduced  to 

(50)  ap2W2=abc. 

From  this  follows  the  theorem  of  Joachimsthal  : 

Along  a  geodesic  or  a  line  of  curvature  on  a  central  quadric  the 
product  of  the  semi-diameter  of  the  quadric  parallel  to  the  tangent 
to  the  curve  at  a  point  P  and  the  distance  from  the  center  to  the 
tangent  plane  at  P  is  constant. 

From  (47)  and  (50)  we  obtain  the  equation 


for  all  points  on  the  quadric.  Since  W  is  the  same  for  all  direc 
tions  at  a  point,  the  maximum  and  minimum  values  of  p  and  R 
correspond.  Hence  we  have  the  theorem  : 

In  the  central  section  of  a  quadric  parallel  to  the  tangent  plane  at 
a  point  P  the  principal  axes  are  parallel  to  the  directions  of  the  lines 
of  curvature  at  P.* 

EXAMPLES 

1.  On  a  hyperbolic  paraboloid,  of  which  the  principal  parabolas  are  equal,  the 
locus  of  a  point,  the  sum  or  difference  of  whose  distances  frotn  the  generators 
through  the  vertex  of  the  paraboloid  is  constant,  is  a  line  of  curvature. 

2.  Find  the  radii  of  curvature  and  torsion,  at  the  extremity  of  the  mean  diam 
eter  of  an  ellipsoid,  of  an  umbilical  geodesic  through  the  pokit. 

3.  Find  the  surfaces  normal  to  the  tangents  to  a  family  of  umbilical  geodesies 
on  an  ellipsoid,  and  determine  the  complementary  surface  (cf.  §  76). 

4.  The  geodesic  distance  of  two  diametrically  opposite  umbilical  points  on  an 
ellipsoid  is  equal  to  one  half  the  length  of  the  principal  section  through  the 
umbilical  points. 

5.  Find  the  form  of  the  linear  element  of  the  hyperboloid  of  two  sheets  or  the 
elliptic  paraboloid,  when  the  parametric  system  is  polar  geodesic  with  an  umbilical 
point  for  pole. 

6.  If  MI  and  M2  are  two  points  of  intersection  of  a  geodesic  through  the  umbilical 
point  0  with  a  line  of  curvature  v  =  const.  ,  then 


tan  —  -  -  cot  -  =  const. 


*  For  a  more  complete  discussion  of  the  geodesies  on  quadrics,  the  reader  is  referred 
to  a  memoir  by  v.  Braunmuhl,  in  Math.  Annalen,  Vol.  XX  (1882),  pp.  556-686. 


EQUATIONS  OF  A  RULED  SURFACE       241 

7.  Given  a  line  of  curvature  on  an  ellipsoid  and  the  geodesies  tangent  to  it; 
the  points  of  intersection  of  pairs  of  these  geodesies,  meeting  orthogonally,  lie  on 
a  sphere. 

8.  Given  the  geodesies  tangent  to  two  lines  of  curvature ;  the  points  of  inter 
section  of  pairs  of  these  geodesies,  meeting  orthogonally,  lie  on  a  sphere. 

103.  Equations  of  a  ruled  surface.  A  surface  which  can  be  gen 
erated  by  the  motion  of  a  straight  line  is  called  a  ruled  surface. 
Developables  are  ruled  surfaces  for  which  the  lines,  called  the 
generators,  are  tangent  to  a  curve.  As  a  general  thing,  ruled  sur 
faces  do  not  possess  this  property,  and  in  this  case  they  are  called 
skew  surfaces.  Now  we  make  a  direct  study  of  ruled  surfaces,  par 
ticularly  those  of  the  skew  type,  limiting  our  discussion  to  the 
case  where  the  generators  are  real.* 

A  ruled  surface  is  completely  determined 
by  a  curve  upon  it  and  the  direction  of  the 
generators  at  their  points  of  meeting  with 
the  curve.  We  call  the  latter  the  directrix          __ 
Z>,  and  the  cone  formed  by  drawing  through 


a  point  lines  parallel  to  the  generators  the  FIG  24 

director-cone.    If  the  coordinates  of  a  point 

MQ  of  D  are  #0,  y^  20,  expressed  in  terms  of  the  arc  v  measured 
from  a  point  of  it,  and  Z,  m,  n  are  the  direction-cosines  of  the  gen 
erator  through  Jf0,  the  equations  of  the  surface  are 

(51)  x  =  x0±lu,         y  =  yQ+mu,          z  =  z0  +  nu, 

where  u  is  the  distance  from  MQ  to  a  point  M  on  the  generator 
through  MQ.  If  00  denotes  the  angle  which  the  generator  through 
MQ  makes  with  the  tangent  at  Jf0  to  Z>,  then 

(52)  cos  00  =  x'J,  +  yrm  +  z'Qn, 

where  the  accent  indicates  differentiation  with  respect  to  v  (fig.  24). 
From  (51)  we  find  for  the  linear  element  the  expression 

(53)  ds*  =du2+2  cos  00  dudv  +  (aV  +  2  bu  +  1)  dv*, 
where  we  have  put  for  the  sake  of  brevity 


*  We  shall  use  the  term  ruled  to  specify  the  surfaces  of  the  skew  type,  and  developable 
for  the  others. 


242  RULED  SURFACES 

Since  the  generators  are  geodesies,  their  orthogonal  trajectories 
can  be  found  by  quadratures  (§  92).  We  arrive  at  this  result 
directly  by  remarking  that  the  equation  of  these  trajectories  is 
(HI,  26)  du  +  cos  60dv  =  0, 

and  that  00  is  a  function  of  v  alone. 

104.  Line  of  striction.  Developable  surfaces.  We  shall  now  con 
sider  the  quantities  which  determine  the  relative  positions  of  the 
generators  of  a  ruled  surface. 

Let  g  and  g'  be  two  generators  determined  by  parametric  values 
v  and  v  +  Sv,  and  let  X,  /*,  v  denote  the  direction-cosines  of  their 
common  perpendicular.     If  the  direction-cosines  of  g  and  g'  be 
denoted  by  I,  m,  n  ;  I  +  SZ,  m  +  8w,  n  +  Bn  respectively,  we  have 
(  l\  +  nifJL  +  nv  =  0, 
|  (I  +  81)  \  +  (m  +  &m)  A*  +  (w  +  Sw)  v  =  0, 
and  consequently 

(56)  \:fji:v  =  (m$n  —  n$m) :  (n&l  —  l&n) :  (ISm  —  mSl). 

From  (54)  it  follows  that 

(mn1-  nm')*  +  (nl'-  ln')*+(lm'-  ml'*       2 
arid  by  Taylor's  theorem, 

(57)  '  l+«  =  l  +  rftr  +  gW+--. 

Hence  equations  (56)  may  be  replaced  by 


(58) 


where  et,  e2,  e3  denote  expressions  of  the  first  and  higher  orders  in  Sv. 
If  Mfa  y,  z)  and  Jf'(z+8s,  y  +  %  2  +  ^2)  are  the  points  of 
meeting  of  this  common  perpendicular  with  #  and  g'  respectively 
(fig.  24),  the  length  MM',  denoted  by  A,  is  given  by 


or 

(60)  A  =  \&x  +  /x%  -f 


LINE  OF  STRICTION  243 

From  (51),  after  the  manner  of  (57),  we  obtain 
Sjc  =  (x'0  -+•  ul')  Bv  -f  IBu  -f-  <7, 

where  cr  involves  the  second  and  higher  powers  of  Bv.    When  this 
and  similar  values  for  By  and  Bz  are  substituted  in  (60),  we  have 

(61)  ^=p  +  €, 

where 


(62) 


I      m    .n 
I'     m1    n' 


and  e  involves  first  and  higher  powers  of  Bv.    In  consequence  of 
(52)  and  (54)  we  have 

(63)  /=««^. 

As  Bv  approaches  zero,  the  point  M  approaches  a  limiting  posi 
tion  C,  which  is  called  the  central  point  of  the  generator.  Let  a 
Denote  the  value  of  u  for  this  point.  In  order  to  find  its  value  we 
remark  that  it  follows  from  the  equations  (55)  and  (59)  that 

Sx  Bl      By  Bm      Bz  Bn  _  ~ 

Bv  Bv      Sv  Bv      Bv  8v 

If  the  above  expressions  for  these  quantities  be  substituted  in  this 
equation,  we  have  in  the  limit,  as  Bv  approaches  zero, 

(64)  a*u  +  b  =  0. 

Consequently 

(65) 


The  locus  of  the  central  points  is  called  the  line  of  striction.  Its 
parametric  equation  is  (64).  Evidently  b  —  0  is  a  necessary  and 
sufficient  condition  that  the  line  of  striction  be  the  directrix. 

From  (61)  and  (63)  it  is  seen  that  the  distance  between  near-by 
generators  is  of  the  second  order  when 

(66)  a2sm26>0-62=:0. 

Without  loss  of  generality  we  may  take  the  line  of  striction 
for  directrix,,  in  which  case  we  may  have  sin#0=:0,  that  is,  the 


244 


KULED  SURFACES 


generators  are  tangent  to  the  directrix.  Another  possibility  is 
afforded  by  a  —  0.  From  (54)  it  is  seen  that  the  only  real  sur 
faces  satisfying  this  condition  are  cylinders.  Hence  (cf.  §  4)  : 

A  necessary  and  sufficient  condition  that  a  ruled  surface,  other  than 
a  cylinder,  be  developable  is  that  the  distance  between  near-by  genera 
tors  be  of  the  second  or  higher  orders  ;  in  this  case  the  edge  of  regres 
sion  is  the  line  of  striction. 

105.  Central  plane.  Parameter  of  distribution.  The  tangent 
plane  to  a  ruled  surface  at  a  point  M  necessarily  contains  the 
generator  through  M.  It  has  been  found  (§  25)  that  for  a  devel 
opable  surface  this  plane  is  tangent  at  all  points  of  the  generator. 

We  shall  see  that  in  the  case  of  skew 
surfaces  the  tangent  plane  varies  as  M 
moves  along  the  generator.  We  deter 
mine  the  character  of  this  variation  by 
finding  the  angle  which  the  tangent 
plane  at  M  makes  with  the  tangent 
plane  at  the  central  point  C  of  the  gen 
erator  through  M.  The  tangent  plane 
at  C  is  called  the  central  plane. 
Let  g  and  gl  be  two  generators,  and  MM  '  their  common  per 
pendicular  (fig.  25).  Through  the  point  M  of  g  draw  the  plane 
normal  to  g  ;  it  meets  g^  in  Mv  and  the  line  through  M  parallel 
to  ffl  in  M2.  The  limiting  positions  of  the  planes  M^MM  and 
M'MM,  as  g^  approaches  g,  are  the  tangent  planes  at  M  and  at  C, 
the  limiting  position  of  M.  The  angle  between  these  planes,  de 
noted  by  (£',  is  equal  to  MMJtt^  and  the  angle  between  g  and  gv 
denoted  by  cr,  is  equal  to  MMM2.  By  construction  MM2Ml  and 
MMM2  are  right  angles.  Hence 

.,      MM.      MM  ton  a- 

tan  d>  =  -  -  =  —  —  • 

MM' 


In  the  limit  M  is  the  central  point  (7,  and  so  we  have 

v  ,,      (u  —  a)da-       ,  (u  —  a)am 

tan<f>=lim  tan<f>'=  -  ^  —  =  ± 


for  we  have 


pdv  p 

da*  =  lim  (SI2  +  8m2  -f  &i2)  =  a  W. 


PARAMETER  OF  DISTRIBUTION  245 

It  is  customary  to  write  the  above  equation  in  the  form 
(67) 


The  function  ft  thus  defined  is  called  the  parameter  of  distribution. 
It  is  the  limit  of  the  ratio  of  the  shortest  distance  between  two 
generators  and  their  included  angle.  As  it  is  independent  of  the 
parameter  u,  we  have  the  theorem : 

The  tangent  of  the  angle  between  the  tangent  plane  to  a  ruled 
surface  at  a  point  M  and  the  central  plane  is  proportional  to  the 
distance  of  M  from  the  central  point. 

From  this  it  follows  that  as  M  moves  along  a  generator  from  —  oo 
to  +  oo,  (/>  varies  from  —  Tr/2  to  7r/2.  Hence  the  tangent  planes  at 
the  infinitely  distant  points  are  perpendicular  to  the  central  plane. 
Since  /3=0  is  the  condition  that  a  surface  be  developable,  the 
tangent  plane  is  the  same  at  all  points  of  the  generator. 

We  shall  now  derive  equation  (67)  analytically.  From  (51)  we  find 
that  the  direction-cosines  of  the  normal  to  the  surface  are  of  the  form 

„     (mz'0  —  ny'Q )  +  (mn'  —  m'n)  u 
(  °)  X=- — 2 — — ^— ; 

the  expressions  for  Y  and  Z  are  similar  to  the  above.  The 
direction-cosines  JT0,  F0,  ZQ  of  the  normal  at  the  central  point  are 
obtained  from  these  by  replacing  u  by  a.  From  this  we  have 

/£\Q\  r\o   x/\  — •  ^T    ~V~V 

_  2  (mz[  —  ny'^f + 2  (mz[  —  ny'Q} (mn'  —  m'n} (u  +  a)  -f  a2ua 

(aV  +  2  bu  +  sin200)*  (a  V  +  2  ba  +  sin200)* 

which  leads  to  4/^2 

^«2J  _.   a  (u  —  a) 


From  this  equation  and  (67)  we  have 


(70)  ._        a0-_ 

(     '  /^~±  2  "       2 


0  0  0 

I      m     n 
'      ' 


When  the  surface  is  defined  by  its  linear  element,  @  is  thus  deter 
mined  only  to  within  an  algebraic  sign.  We  shall  find,  however, 
that  this  is  not  the  case  when  the  surface  is  defined  by  equations 
of  the  form  (51). 


246  EJJLED  SURFACES 

To  this  end  we  take  a  particular  generator  g  for  the  z-axis. 
Then  for  g  we  have 

Let  also  the  central  plane  be  taken  for  the  zz-plane  and  the  central 
point  for  the  origin.  From  (68)  it  follows  that  y'Q  =  0.  Since  the 
origin  is  the  central  point,  b  =  0  and  consequently  I'  =  0.  Hence 
the  equation  of  the  tangent  plane  at  a  point  of  g  has  the  simple  form 

(71)  m'u%  —  X'OT)  =  0, 

f  and  rj  being  current  coordinates.  If  the  coordinate  axes  have 
the  usual  orientation,  and  the  angle  *  is  measured  positively  in 
the  direction  from  the  positive  #-axis  to  the  positive  ^/-axis, 

from  equation  (71)  we  have 

m'u 

(7  2)  tan  *  =  — —  • 

Comparing  this  with  equation  (67),  we  find  for  ft  the  value  x'Jm'. 
In  order  to  obtain  the  same  value  from  (70)  for  these  particular 
values,  we  must  take  the  negative  sign.  Hence  we  have,  in 
general,  \  /  / 

(73)  0  =  -i    I      m     n 

2     I'     m1    n' 

It  is  seen  from  (72)  that,  as  a  point  moves  along  a  generator  in 
the  direction  of  u  increasing,  the  motion  of  the  tangent  plane  is 
that  of  a  right-handed  or  left-handed  screw,  according  as  ft  is 
negative  or  positive. 

EXAMPLES 

1 .  Show  that  for  the  ruled  surface  denned  by 

2  J  2 

_  i  r ,-  .  .  ^  .  -,    .  * 

y~2> 


=  Cu<t>du 


where  0  and  \f/  are  any  functions  of  w,  the  directrix  and  the  generators  are  minimal. 
Determine  under  what  condition  the  curvature  of  the  surface  is  constant. 

2.  Determine  the  condition  that  the  directrix  of  a  ruled  surface  be  a  geodesic. 


PARAMETER  OF  DISTRIBUTION  247 

3.  Prove,  by  means  of  (62),  that  the  lines  of  curvature  of  a  surface  F(x,  y,  z)  =  0 
are  defined  by  ^        dy,         dz 

dF         d_F         cF 
dx          dy          dz 

>*,   a**,   *?* 

dx          dy          dz 

4.  The  right  helicoid  is  the  only  ruled  surface  whose  generators  are  the  principal 
normals  of  their  orthogonal  trajectories.    Find  the  parameter  of  distribution. 

5  .  Prove  for  the  hyperboloid  of  revolution  of  one  sheet  that  : 

(a)  the  minimum  circle  is  the  line  of  striction  and  a  geodesic  ; 
(6)  the  parameter  of  distribution  is  constant. 

6.  With  every  point  P  on  a  ruled  surface  there  is  associated  another  point  P' 
on  the  same  generator,  such  that  the  tangent  planes  at  these  points  are  perpendicular. 
Prove  that  the  product  OP  •  OP',  where  0  denotes  the  central  point,  has  the  same 
value  for  all  points  P  on  the  same  generator. 

7.  The  normals  to  a  ruled  surface  along  a  generator  form  a  hyperbolic  paraboloid. 

8.  The  cross-ratio  of  four  tangent  planes  to  a  ruled  surface  at  points  of  a  gen 
erator  is  equal  to  the  cross-ratio  of  the  points. 

9.  If  two  ruled  surfaces  are  symmetric  with  respect  to  a  plane,  the  values  of 
the  parameter  of  distribution  for  homologous  generators  differ  only  in  sign. 

106.  Particular  form  of  the  linear  element.  A  number  of  prop 
erties  of  ruled  surfaces  are  readily  obtained  when  the  linear  element 
is  given  a  particular  form,  which  we  will  now  deduce. 

Let  an  orthogonal  trajectory  of  the  generators  be  taken  for  the 
directrix.  In  this  case 

(74)  *,=,       f,-*.  » 


If  we  make  the  change  of  parameters, 

Cv 

(75)  u  =  u,         vl  =  I   a  dv, 

Jo 

the  linear  element  (53)  is  reducible  to 

(76)  ds*  =  du*  +  [(u  -  a)2  +  /32]  dv*. 

The  angle  6  which  a  curve  vl=f(u)  makes  with  the  generators  is 
given  by 


(77)  tan0  =  V(w  — 

Also  the  expression  for  the  total  curvature  is 

(78)  JT  =  -- f* 


248  KULED  SURFACES 

Hence  a  real  ruled  surface  has  no  elliptic  points.  All  the  points 
are  hyperbolic  except  along  the  generators  for  which  /3  =  0,  and 
at  the  infinitely  distant  points  on  each  generator.  Consequently 
the  linear  element  of  a  developable  surface  may  be  put  in  the  form 

(79)  ds*  =  du"  +  (u  -  a)'2dv*. 

Also,  in  the  region  of  the  infinitely  distant  points  of  a  ruled  sur 
face  the  latter  has  the  character  of  a  developable  surface.  As 
another  consequence  of  (78)  we  have  that,  for  the  points  of  a 
generator  the  curvature  is  greatest  in  absolute  value  at  the  cen 
tral  point,  and  that  at  points  equally  distant  from  the  latter  it 
has  the  same  value. 

When  the  linear  element  is  in  the  form  (76),  the  Gauss  equation 
of  geodesies  (VI,  56)  has  the  form 


V(M  -  a)'2  +  @*d6  +  (u-a) dv1  =  0. 

An  immediate  consequence  is  the  theorem  of  Bonnet : 

If  a  curve  upon  a  ruled  surface  has  two  of  the  following  properties, 
it  has  the  third  also,  namely  that  it  cut  the  generators  under  constant 
angle,  that  it  be  a  geodesic  and  that  it  be  the  line  of  striction. 

A  surface  of  this  kind  is  formed  by  the  family  of  straight  lines 
which  cut  a  twisted  curve  under  constant  angle  and  are  perpen 
dicular  to  its  principal  normals.  A  particular  case  is  the  surface 
formed  of  the  binomials  of  a  curve.  It  is  readily  shown  from  (73) 
that  the  parameter  of  distribution  of  this  surface  is  equal  to  the 
radius  of  torsion  of  the  curve.  ,-' 

107.  Asymptotic  lines.  Orthogonal  parametric  systems.  The  gen 
erators  are  necessarily  asymptotic  lines  on  a  ruled  surface.  We 
consider  now  the  other  family  of  these  lines.  From  (51)  and  (68) 


we  find  r 

1     ' 


(80)       i>  =  0,  D'=- 


I     m     n 


H     '       '       '  "    z"+n"u    n    z'  +  n'u 


m 


Hence   the  differential   equation  of  the   other  family  of  asymp 
totic  lines  is  of  the  form 

— 

dv 


ASYMPTOTIC  LINES  249 

where  £,  Jf,  N  are  functions  of  v.    As  this  is  an  equation  of  the 
Riccati  t}^pe,  we  have,  from  §  14,  the  theorem  of  Serret: 

The  four  points  in  which  each  generator  of  a  ruled  surface  is  cut 
by  four  curved  asymptotic  lines  are  in  constant  cross-ratio. 

From  §  14  it  follows  also  that  when  one  of  these  asymptotic 
lines  is  known  the  others  can  be  found  by  quadratures. 

When  the  surface  is  referred  to  an  orthogonal  system  and  the  linear  element  is 
in  the  form  (76),  written 
(81)  ds2  =  du2  +  a2  [(u  -  a)2  +  /32]  dv2, 

the  expressions  (80)  can  be  given  a  simpler  form. 
From  (73)  and  (81)  we  have 


From  the  equations        'Lx'ol  =  0,      Sz62  =  1,  2Z2  =  1, 

and  (54)  we  obtain,  by  differentiation, 

Zzfceo'  =  0,        ZM'  =  0,          Zatf'J  =  -b, 

-Ll'l"  =  aa',    ZK"  =  -  a2,    ZJ'x6'  =  &'-«, 
where  t  is  defined  by  zr'«6  =  t. 

If  the  expression  for  D"  in  (80)  be  multiplied  by  the  determinant  of  the  right- 
hand  member  of  (73),  and  the  result  be  divided  by  its  equal,  —  a2/3,  we  have,  in 
consequence  of  the  above  identities, 

D"  =  -  —  i—  [w2  (to?  -  aa'fc)  +  u  (2  tb  -  aa'  -  66')  +  t  -  &']. 
2 


If  equations  (74)  be  solved  for  a  and  6  as  functions  of  a  and  0,  and  the  resulting 
expressions  be  substituted  in  this  equation,  we  have 

D"  =  -~{r[(u  -  a)2  +  ^]  +  ?(u  -  a)  +  /3a'}, 

where  the  primes  indicate  differentiation  with  respect  to  Vi,  given  by  (75),  and  r 
is  defined  by  „/ 


From  the  above  equations  it  follows  that  the  mean  curvature  (cf  .  §  52)  is  express 
ible  in  the  form 

(82)  J  -  +  *  =  -    r  ^u  ~  a)2  +  ^  + 

Pi      Pz  [(u  -  a-)2 


EXAMPLES 

1.  When  the  linear  element  of  a  ruled  surface  is  in  the  form  (76),  the  direction- 
cosines  of  the  limiting  position  of  the  common  perpendicular  to  two  generators  are 

?<L±£?,         V*  +  <™'t        z'«  +  «n' 
«/3  aft  o/3 


250  MINIMAL  SURFACES 

2.  Prove  that  the  developable  surfaces  are  the  only  ruled  surfaces  with  real 
generators  whose  total  curvature  is  constant. 

3.  Show  that  the  perpendicular  upon  the  z-axis  from  any  point  of  the  cubic 
x  _  u^y  —  M2?  z  —  wa  lies  in  the  osculating  plane  at  the  point,  and  lind  the  asymp 
totic  lines  on  the  ruled  surface  generated  by  this  perpendicular. 

4.  Determine  the  function  0  in  the  equations 

x  =  w,         y  =  un,        z  =  0(u), 

so  that  the  osculating  plane  at  any  point  M  of  this  curve  shall  pass  through  the 
projection  P  of  M  on  the  y-axis.  Find  the  asymptotic  lines  on  the  surface  gener 
ated  by  the  line  MP. 

5.  Show  that  the  equations 

x  =  M  sin  0  cos  ^,        y  =  wsinflsin^,        z  =  u-fwcos0, 

where  6  and  ^  are  functions  of  t>,  define  the  most  general  ruled  surface  with  a  rec 
tilinear  directrix,  and  prove  that  the  equation  of  asymptotic  lines  can  be  integrated 
by  two  quadratures.  Discuss  the  case  where  0  is  constant. 

6.  Concerning  the  curved  asymptotic  lines  on  a  ruled  surface  the  following  are 
to  be  proved  : 

(a)  if  one  of  them  is  an  orthogonal  trajectory  of  the  generators,  the  determina 
tion  of  the  rest  reduces  to  quadratures  ; 

(6)  if  two  of  them  are  orthogonal  trajectories,  they  are  curves  of  Bertrand  ; 
(c)  if  all  of  them  are  orthogonal  trajectories,  the  surface  is  a  right  helicoid. 

7.  Determine  the  condition  that  the  line  of  striction  be  an  asymptotic  line,  and 
show  that  in  this  case  the  other  curved  asymptotic  lines  can  be  found  by  quadratures. 

8.  Find  a  ruled  surface  of  the  fourth  degree  which  is  generated  by  a  line  pass 
ing  through  the  two  lines  x  =  0,  y  =  0  ;  z  =  0,«  +  y-fz  =  l.    Show  that  these  lines 
and  the  line  x  =  0,  x  +  y  +  z  =  1  are  double  lines.    Find  the  line  of  striction. 

9.  The  right  helicoid  is  the  only  ruled  surface  each  of  whose  lines  of  curvature 
cuts  the  generators  under  constant  angle  ;  however,  on  any  other  ruled  surface 
there  are  in  general  four  lines  of  curvature  which  have  this  property. 

108.  Minimal  surfaces.  In  1760  Lagrange  extended  to  double 
integrals  the  Euler  theorems  about  simple  integrals  in  the  calculus 
of  variations,  and  as  an  example  he  proposed  the  following  problem  *  : 

Given  a  closed  curve  C  and  a  connected  surface  S  bounded  by  the 
curve;  to  determine  8  so  that  the  inclosed  area  shall  be  a  minimum. 

If  the  surface  be  denned  by  the  equation 

z  =f(x,  y), 

the  problem  requires  the  determination  of  f(x,  y)  so  that  the  inte 
gral  (cf.  Ex.  1,  p.  77) 


*  CEuvres  de  Lagrange,  Vol.  I,  pp.  354-357.   Paris,  1867. 


MINIMAL    SURFACES  251 

extended  over  the  portion  of  the  surface  bounded  by  C  shall  be  a 
minimum.  As  shown  by  Lagrange,  the  condition  for  this  is 

(83) 

or,  in  other  form, 

(84)  (1  +  qz)r  -  2pqs  +  (1  +  p*)t  =  0. 

Lagrange  left  the  solution  of  the  problem  in  this  form,  and 
Meusnier,*  sixteen  years  later,  proved  that  this  equation  is 
equivalent  to  the  vanishing  of  the  mean  curvature  (§  52),  thus 
showing  that  the  surfaces  furnishing  the  solution  of  Lagrange's 
problem  are  characterized  by  the  geometrical  property  which  now 
is  usually  taken  as  the  definition  of  minimal  surfaces;  however,  the 
name  indicates  the  connection  with  the  definition  of  Lagrange. f 

In  what  follows  we  purpose  giving  a  discussion  of  minimal  sur 
faces  from  the  standpoint  of  their  definition  as  the  surfaces  whose 
mean  curvature  is  zero  at  all  the  points.  At  each  point  of  such  a 
surface  the  principal  radii  differ  only  in  sign,  and  so  every  point 
is  a  hyperbolic  point  and  its  Dupin  indicatrix  is  an  equilateral 
hyperbola.  Consequently  minimal  surfaces  are  characterized  by 
the  property  that  their  asymptotic  lines  form  an  orthogonal  sys 
tem.  Moreover,  the  tangents  to  the  two  asymptotic  lines  at  a 
point  bisect  the  angles  between  the  lines  of  curvature  at  the  point, 
and  vice  versa. 

We  recall  the  formulas  giving  the  relations  between  the  funda 
mental  quantities  of  a  surface  and  its  spherical  representation 
(IV,  70) : 

(85)  (o  = 


From  these  we  have  at  once  the  theorem : 

The  necessary  and  sufficient  condition  that  the  spherical  represen 
tation  of  a  surface  be  conformal  is  that  the  surface  be  minimal. 

*  Memoire  sur  la  courbure  des  surfaces,  Memoires  des  Savants  Strangers,  Vol.  X 
(1785),  p.  477. 

t  For  a  historical  sketch  of  the  development  of  the  theory  of  minimal  surfaces  and  a 
complete  discussion  of  them  the  reader  is  referred  to  the  Lemons  of  Darboux  (Vol.  I,  pp. 
267  et  seq.).  The  questions  in  the  calculus  of  variations  involved  in  the  study  of  mini 
mal  surfaces  are  treated  by  Riemann,  Gesammelte  Werke,  p.  287  (Leipzig,  1876) ;  and  by 
Schwarz,  Gesammelte  Abhandlungen,  Vol.  I,  pp.  223,  270  (Berlin,  1890). 


252  MINIMAL  SURFACES 

Hence  isothermal  orthogonal  systems  on  the  surface  are  repre 
sented  by  similar  systems  on  the  sphere,  and  conversely.  All  the 
isothermal  orthogonal  systems  on  the  sphere  are  known  (§§  35,  40). 
Suppose  that  one  of  these  systems  is  parametric  and  that  the  linear 
element  is  * 


From  the  general  condition  for  minimal  surfaces  (IV,  77),  namely 

(86)  <£D"  +  3D  -  2  &Df  =  0, 
it  follows  that  in  this  case  n1'  —  0 

In  consequence  of  this  the  Codazzi  equations  (V,  27)  are  reducible  to 

(87)  ?-^-0,         f  +  ~?'  =  °- 

dv       du  du        dv 

By  eliminating  D  or  D'  we  find  that  both  D  and  D1  are  integrals 
of  the  equation  ~IQ  ^Q 

a?  +  ^?=  °* 

Hence  the  most  general  form  of  D'  is 

(88)  D'  =  $  (u  +  iv)  +  ^(u-  iv), 

where  <f>  and  i/r  are  arbitrary  functions.    Then  from  (87)  we  have 

(89)  D  =  ~D"  =  -  i((j)  -ty+c, 

where  c  is  the  constant  of  integration.  To  each  pair  of  functions 
<£,  T/T  there  corresponds  a  minimal  surface  whose  Cartesian  coordi 
nates  are  given  by  the  quadratures  (V,  26),  namely 


(90)  =  - 

'       du          \\     du  dv  dv          \        du  dv 

and  similar  expressions  in  y  and  z.  Evidently  the  surface  is  real 
only  when  <f>  and  i/r  are  conjugate  functions. 

In  obtaining  the  preceding  results  we  have  tacitly  assumed  that 
neither  D  nor  D1  is  zero.  We  notice  that  either  may  be  zero  and 
then  the  other  is  a  constant,  which  is  zero  only  for  the  plane. 
These  results  may  be  stated  thus: 

Every  isothermal  system  on  the  sphere  is  the  representation  of  the 
lines  of  curvature  of  a  unique  minimal  surface  and  of  the  asymptotic 
lines  of  another  minimal  surface. 


LINES  OF  CURVATURE  AND  ASYMPTOTIC  LINES    253 

The  converse  also  is  true,  namely: 

The  spherical  representations  of  the  lines  of  curvature  and  of  the 
asymptotic  lines  of  a  minimal  surface  are  isothermal  systems. 

For,  if  the  lines  of  curvature  are  parametric,  equation  (86)  may 
be  replaced  by  D  =  p^         D>,  =  _  pg 

where  p  is  equal  to  either  principal  radius  to  within  its  algebraic 
sign.  When  these  values  and  D'  =  &  =  0  are  substituted  in  the 
Codazzi  equations  (V,  27),  we  obtain 


so  that  £/g=*U/V,  which  proves  the  first  part  of  the  theorem  (§  41). 
When  the  asymptotic  lines  are  parametric,  we  have  Z>=D"=c^=0, 
and  equations  (V,  27)  reduce  to 


(>!")=»• 


cu 
from  which  it  follows  that  <£~/^=  U/V. 

109.  Lines  of  curvature  and  asymptotic  lines.  Adjoint  minimal 
surfaces.  We  return  to  the  consideration  of  equations  (87)  and 
investigate  first  the  minimal  surface  with  its  lines  of  curvature 
represented  by  an  isothermal  system.  Without  loss  of  generality,* 
we  may  take 

(91)  D  =  -D"  =  1,         £>'=0. 

From  (IV,  77)  it  follows  that 

—  =  --2  =  ~X2,          E=G  =  p, 

PiP2         Pi 
where  />  =  |^|  =  |p2J. 

Hence  we  have  the  theorem  : 

The  parameters  of  the  lines  of  curvature  of  a  minimal  surface  may 
be  so  chosen  that  the  linear  elements  of  the  surface  and  of  its  spher 
ical  representation  have  the  respective  forms 

ds2  =  p  (du2  +  dv2),         dd2  =  -  (du2  +  dv2), 

P 

where  p  is  the  absolute  value  of  each  principal  radius. 

*  Any  other  value  of  the  constant  leads  to  homothetic  surfaces. 


254  MINIMAL  SURFACES 

In  like  manner  we  may  take,  for  the  solution  of  equations  (87), 
(92)  D  =  I>"=Q,  D'  =  l. 

Again  we  find       ..  -, 

J-  =  -±  =  -\\          E=-G  =  p, 
PiP*          Pi 
so  that  we  have  a  result  similar  to  the  above  : 

The  parameters  of  the  asymptotic  lines  of  a  minimal  surface  may 
be  so  chosen  that  the  linear  elements  of  the  surface  and  of  its  spherical 
representation  have  the  respective  forms 

ds*  =  p  (du*  +  di?),         d<r*  =  -  (du*  +  dv2), 

where  p  is  the  absolute  value  of  each  principal  radius. 

From  the  symmetric  form  of  equations  (87)  it  follows  that  if 
(88)  and  (89)  represent  one  set  of  solutions,  another  set  is  given  by 


These  values  are  such  that 


which  is  the  condition  that  asymptotic  lines  on  either  surface  cor 
respond  to  a  conjugate  system  on  the  other  (§  56).  When  this 
condition  is  satisfied  by  two  minimal  surfaces,  and  the  tangent 
planes  at  corresponding  points  are  parallel,  the  two  surfaces  are 
said  to  be  the  adjoints  of  one  another.  Hence  a  pair  of  functions 
<£,  ->/r  determines  a  pair  of  adjoint  minimal  surfaces.  When,  in  par 
ticular,  the  asymptotic  lines  on  one  surface  a*e  parametric,  the 
functions  have  the  values  (92),  and  on  the  other  the  values  (91). 
It  follows,  then,  from  (90),  that  between  the  Cartesian  coordinates 
of  a  minimal  surface  and  its  adjoint  the  following  relations  hold: 

cjx\_  _dx  foi=fa. 

cu          dv'         dv      du 

and  similar  expressions  in  the  «/'s  and  z's,  when  the  parametric 
curves  are  asymptotic  on  the  locus  of  (#,  #,  z). 

110.  Minimal  curves  on  a  minimal  surface.  The  lines  of  length 
zero  upon  a  minimal  surface  are  of  fundamental  importance.  When 
they  are  taken  for  parametric  curves,  the  equations  of  the  surface 
take  a  simple  form,  which  we  shall  now  obtain. 


MINIMAL  CURVES  255 

Since  the  lines  of  length  zero,  or  minimal  lines,  are  parametric, 
we  have 

(94)  ^  =  £  =  0. 

From  (85)  it  follows  that  the  parametric  lines  on  the  sphere  also  are 
minimal  lines,  that  is,  the  imaginary  rectilinear  generators.  And 
from  (86)  we  find  that  1)'  is  zero.  Conversely,  when  the  latter  is 
zero,  and  the  parametric  lines  are  minimal  curves,  it  follows  from 
(IV,  33)  that  Km  is  equal  to  zero.  Hence : 

A  necessary  and  sufficient  condition  that  a  surface  be  minimal  is 
that  the  lines  of  length  zero  form  a  conjugate  system.* 

In  consequence  of  (94)  and  (VI,  26)  the  point  equation  of  a 
minimal  surface,  referred  to  its  minimal  lines,  is 


ducv 
Hence  the  finite  equations  of  the  surface  are  of  the  form 

where   U^   T/2,   Us  are   functions   of   u   alone,    and   Fx,   F2,   F3   are 
functions  of  v  alone,  satisfying  the  conditions 


(96)  U?  +  V?  +  U?  =  0,          F{2  +  Fia  +  Fj2  =  0. 

From  (95)  it  is  seen  that  minimal  surfaces  are  surfaces  of  trans 
lation  (§  81),  and  from  (96)  that  the  generators  are  minimal 
curves  (§  22).  In  consequence  of  the  second  theorem  of  §  81  we 
may  state  this  result  thus : 

A  minimal  surface  is  the  locus  of  the  mid-points  of  the  joins  of 
points  on  two  minimal  curves. 

In  §  22  we  found  that  the  Cartesian  coordinates  of  any  minimal 
curve  are  expressible  in  the  form 

(97)  f  (1  -  u*)F(u)  du,      i  f  (1  +  u2)  F(u)  du,      2  Cu  F(u)  du. 

*This  follows  also  from  the  fact  that  an  equilateral  hyperbola  is  the  only  conic  for 
which  the  directions  with  angular  coefficients  ±  i  are  conjugate. 


256  MINIMAL  SURFACES 

Hence  by  the   above   theorem  the   following   equations,  due  to 
Enneper  *,  define  a  minimal  surface  referred  to  its  minimal  lines : 


(98) 


z=  I  u F(u) du  +  I  v&(v) dv, 


where  F  and  4>  are  any  analytic  functions  whatever.  Moreover, 
any  minimal  surface  can  be  defined  by  equations  of  this  form. 
For,  the  only  apparent  lack  of  generality  is  due  to  the  fact  that 
the  algebraic  signs  of  the  expressions  (98)  are  not  determined 
by  equations  (96),  and  consequently  the  signs  preceding  the 
terms  in  the  right-hand  members  of  equations  (98)  could  be 
positive  or  negative.  But  it  can  be  shown  that  by  a  suitable 
change  of  the  parameters  and  of  the  functions  F  and  3>  all  of 
these  cases  reduce  to  (98).  Thus,  for  example,  we  consider  the 
surface  defined  by  the  equations  which  result  when  the  second 
terms  of  the  right-hand  members  of  (98)  are  replaced  by 


In  order  that  the  surface  thus  defined  can  be  brought  into  coin 
cidence,  by  a  translation,  with  the  surface  (98),  we  must  have 


Dividing  these  equations,  member  by  member,  we  have 

from  which  it  follows  that 

Substituting  this  value  in  the  last  of  the  above  equations,  we  find 

*  Zeitschrift  fur  Mathematik  und  Physik,  Vol.  IX  (1864),  p.  107. 


MINIMAL  CUEVES  257 

and  this  value  satisfies  the  other  equations.  Similar  results  fol 
low  when  another  choice  of  signs  is  made.  The  reason  for  the 
particular  choice  made  in  (98)  will  be  seen  when  we  discuss  the 
reality  of  the  surfaces. 

Incidentally  we  have  proved  the  theorem  : 

When  a  minimal  surface  is  defined  by  equations  (98),  the  necessary 
and  sufficient  condition  that  the  two  generating  curves  be  congruent 
is  that 

(99)  ,(«)._1 

From  (98)  we  obtain 


so  that  the  linear  element  is 

(100)  ds2  =  (l  +  uv}*F(u)3>(v)dudv. 

We  find  for  the  expressions  of  the  direction-cosines  of  the  normal 


1  H-  uv  1  +  uv  1  +  uv 

and  the  linear  element  of  the  sphere  is 

,2       4  dudv 

d<r  —  —  -  -  > 

Alt,  (1  +  ««») 

Also  we  have 


(102)  D  = 

so  that  the  equations  of  the  lines  of  curvature  and  of  the  asymp 
totic  lines  are  respectively 

(103)  F(u)  du2  -  3>  (v)  dv*  =  0, 

(104)  F(u)  du2  +  <$>  (v)  dv*  =  0. 

These  equations  are  of  such  a  form  that  we  have  the  theorem  : 

When  a  minimal  surface  is  referred  to  its  minimal  lines,  the  finite 
equations  of  the  lines  of  curvature  and  asymptotic  lines  are  given  by 
quadratures,  which  are  the  same  in  both  cases. 

In  order  that  a  surface  be  real  its  spherical  representation  must 
be  real.  Consequently  u  and  v  must  be  conjugate  imaginaries,  as 


258  MINIMAL  SURFACES 

is  seen  from  (101)  and  §  13,  and  the  functions  F  and  <£  must  be 
conjugate  imaginary.  Hence  if  RO  denotes  the  real  part  of  a 
function  9,  all  real  minimal  surfaces  are  defined  by 

x  =  fi  f  (1  -  u2)  F(u)  du,         y  =  R  ft  (1  +  u2)  F(u)  du, 
z=R  I  2uF(u)du, 

where  F(u)  is  any  function  whatever  of  a  complex  variable  u. 
In  like  manner  the  equations  of  the  lines  of  curvature  may  be 
written  in  the  form 

(105)  72  /  ^/F(u)du  =  const.,       11  \  iVF(u)du  =  const. 

111.  Double  minimal  surfaces.  It  is  natural  to  inquire  whether 
the  same  minimal  surface  can  be  denned  in  more  than  one  way  by 
equations  of  the  form  (98).  We  assume  that  this  is  possible,  and 
indicate  by  uv  v^  and  F^(u^  ^V^)  the  corresponding  parameters 
and  functions.  As  the  parameters  u^  vl  refer  to  the  lines  of  length 
zero  on  the  surface,  each  is  a  function  of  either  u  or  v.  In  order  to 
determine  the  forms  of  the  latter  we  make  use  of  the  fact  that  the 
positive  directions  of  the  normal  to  the  surface  in  the  two  forms  of 
parametric  representation  may  have  the  same  or  opposite  senses. 
When  they  have  the  same  sense,  the  expressions  (101)  and  similar 
ones  in  uv  and  vl  must  be  equal  respectively.  In  this  case 

(106)  %!=!*,  v^v. 

If  the  senses  are  opposite,  the  respective  expressions  are  equal  to 
within  algebraic  signs.  From  the  resulting  equations  we  find 


(107) 


u 


When  we  compare  equations  (98)  with  analogous  equations  in 
ul  and  vv  we  find  that  for  the  case  (106)  we  must  have 


and  for  the  case  (107) 


DOUBLE  MINIMAL  SUBFACES  259 

Hence  we  have  the  theorem  : 

A  necessary  and  sufficient  condition  that  two  minimal  surfaces,  deter 
mined  by  the  pairs  of  functions  F,  <&  and  Fv  <&v  be  congruent  is  that 


(108)        ^w— 

to  the  point  (u,  v)  on  one  surface  corresponds  the  point  (  --  »  -- 


on 
u 


the  other,  and  the  normals  at  these  points  are  parallel  but  of  different 

sense. 

In  general,  the  functions  F  and  Fl  as  given  by  (108)  are  not  the 
same.  If  they  are,  so  also  are  <£  and  4>r  Ih  this  case  the  right-hand 
members  of  equations  (98)  are  unaltered  when  u  and  v  are  replaced 
by  —  l/v  and  —1/u  respectively.  Hence  the  Cartesian  coordinates 

of  the  points  (u,  v)  and  (  --  >  --  j  differ  at  most  by  constants.   And 

so  the  regions  of  the  surface  about  these  points  either  coincide  or 
can  be  brought  into  coincidence  by  a  translation.  In  the  latter  case 
the  surface  is  periodic  and  consequently  transcendental. 

Suppose  that  it  is  not  periodic,  and  consider  a  point  -ZjJ(w0,  VQ).  As 
u  varies  continuously  from  UQ  to  —  l/t>0,  v  varies  from  v0  to  —  l/w0, 
and  the  point  describes  a  closed  curve  on  the  surface  by  returning 
to  PQ.  But  now  the  positive  normal  is  on  the  other  side  of  the  sur 
face.  Hence  these  surfaces  have  the  property  that  a  point  can  pass 
continuously  from  one  side  to  the  other  without  going  through  the 
surface.  On  this  account  they  were  called  double  minimal  surfaces 
by  Lie,*  who  was  the  first  to  study  them. 

From  the  third  theorem  of  §  110  it  follows  that  double  minimal 
surfaces  are  characterized  by  the  property  that  the  minimal  curves 
in  both  systems  are  congruent.  The  equations  of  such  a  surface 
may  be  written 


The  surface  is  consequently  the  locus  of  the  mid-points  of  the 
chords  of  the  curve 

f  =/,(«),     i  =/,(«),     ?=/,(«), 

which  lies  upon  the  surface  and  is  the  envelope  of  the  parametric 
curves. 

*  Math.  Annalen,  Vol.  XIV  (1878),  pp.  345-350. 


260  MINIMAL  SURFACES 


EXAMPLES 

1.  The  focal  sheets  of  a  minimal  surface  are  applicable  to  one  another  and  to 
the  surface  of  revolution  of  the  evolute  of  the  catenary  about  the  axis  of  the  latter. 

2.  Show  that  there  are  no  minimal  surfaces  with  the  minimal  lines  in  one 
family  straight. 

3.  If  two  minimal  surfaces  correspond  with  parallelism  of  tangent  planes,  the 
minimal  curves  on  the  two  surfaces  correspond. 

4.  If  two  minimal  surfaces  correspond  with  parallelism  of  tangent  planes,  and 
the  joins  of  corresponding  points  be  divided  in  the  same  ratio,  the  locus  of  the 
points  of  division  is  a  minimal  surface. 

5.  Show  that  the  right  helicoid  is  defined  by  F(u)  =  im/2  w2,  where  m  is  a  real 
constant,  and  that  it  is  a  double  surface. 

2 

6.  The  surface  for  which  F(u)  =  -  is  called  the  surface  of  Scherk.   Find  its 

equation  in  the  Monge  form  z  =  f(x,  y).    Show  that  it  is  doubly  periodic  and  that 
it  is  a  surface  of  translation  with  real  generators  which  are  in  perpendicular  planes. 

7.  By  definition  a  meridian  curve  on  a  surface  is  one  whose  spherical  representa 
tion  is  a  great  circle  on  the  unit  sphere.    Show  that  the  surface  of  Scherk  possesses 
two  families  of  plane  meridian  curves. 

112.  Algebraic  minimal  surfaces.  Weierstrass  *  remarked  that 
formulas  (98)  can  be  put  in  a  form  free  of  all  quadratures.  This 
is  done  by  replacing  F(u)  and  <J>(v)  by  f'"(u)  and  #'"(v),  where 
the  accents  indicate  differentiation,  and  then  integrating  by 
parts.  This  gives 


(109) 


x  p. 


2 


+  uf'(u)  -f(u)  +  —^-  4>"(v)  +  v$(v) 
•"(u)  —  iuf(u)  +  if(u)  —  i  -  -^-  <l>"(v)  -h  iv( 


=  uf"(u)  -»)  +  v<f>"(v)  -  4>'(v). 

It  is  clear  that  the  surface  so  denned  is  real  when  /  and  <f>  are 
conjugate  imaginary  functions.  In  this  case  the  above  formulas 
may  be  written  : 

=  R[(l-  u*)f"(u)  +  2  uf'(u)  - 
(110)  y  =  Ri  [(1  +  u*)f"(u)  -  2  uf'(u)  + 


*  Monatsberichte  der  Berliner  Akademie  (1866),  p.  619. 


ALGEBRAIC  MINIMAL  SURFACES  261 

However,  it  is  not  necessary,  as  Darboux  *  has  pointed  out,  that 
f  and  (f>  be  conjugate  imaginaries  in  order  that  the  surface  be  real. 
For,  equations  (109)  are  unaltered  if  /and  0  be  replaced  by 

ft(u)  =f(u)  +  A  (1  -  u2)  +  Bi  (1  +  u*)  +  2  Cu, 
^(v)  =  (f>(v)  -  A(l  -  v2)  +  Bi  ({.-{-  v2)-  2  Cv, 

where  A,  B,  C  are  any  constants  whatever.  Evidently,  if  /  and  </> 
are  conjugate  imaginaries,  the  same  is  not  true  in  general  of  /,_ 
and  <f>l  ;  but  the  surface  was  real  for  the  former  and  consequently 
is  real  for  the  latter  also.  It  is  readily  found  that  /t  and  </>x 
are  conjugate  imaginary  functions  only  in  case  J,  J5,  C  are  pure 
imaginaries. 

Formulas  (109)  are  of  particular  value  in  the  study  of  algebraic 
surfaces.  Thus,  it  is  evident  that  the  surface  is  algebraic  when/ 
and  <£  are  algebraic.  Conversely,  every  algebraic  minimal  surface 
is  determined  by  algebraic  functions  /  and  </>.  In  proving  this  we 
follow  the  method  suggested  by  Weierstrass.f 

We  establish  first  the  following  lemma  : 

Gttven  a  function  $*(?  +  *??)  and  let  "^(f,  77)  denote  the  real  part 
of  4>  ;  if  in  a  certain  domain  an  algebraic  relation  exists  between  M*, 
£,  and  77,  4>  is  an  algebraic  function  of  j~  +  irj. 

If  the  point  f  =  0,  rj  =  0  does  not  lie  within  the  domain  under 
consideration,  this  can  be  effected  by  a  change  of  variables  without 
vitiating  the  argument.  Assuming  that  this  has  been  done,  we 
develop  the  function  <E>  in  a  power  series,  thus  : 

4>  =  a0  +  #o  +  K  +  ibj  (|  +  irj)  +  (a2  +  ib2)  (f  +  irj)'2  +  .  .  .  , 
where  the  a's  and  5's  are  real  constants.    Evidently  M*  is  given  by 


1  (a,  -  ^)  (f  -  ^)  +  J  (a,  -  i68)  (f  -  ii?)2  +  -  -  -  . 

Let  J^(^,  f  ,  ?;)  =  0  denote  a  rational  integral  relation  between  "SP, 
f,  and  77.  When  M*  has  been  replaced  by  the  above  value,  and  the 
resulting  expression  is  arranged  in  powers  of  |  and  77,  the  coeffi 
cient  of  every  term  is  identically  zero.  They  will  continue  to  be 
zero  when  f  and  77  have  been  replaced  by  two  complex  quantities 

*  Vol.  I,  p.  293.          f  Monatsberichte  tier  Berliner  Akademie  (1867),  pp.  511-518. 


262  MINIMAL  SURFACES 

a  and  /3,  provided  that  the  development  remains  convergent.  The 
condition  for  the  latter  is  that  the  moduli  of  a  and  /3  be  each  one 
half  the  modulus  of  f  +  irj.  This  condition  is  satisfied  if  we  take 


Now  we  have 


-•  -. 

-  K-  i50)  +  -  <£  (f 


t'irt,  f  +  ti,]  =  0, 


which  proves  the  lemma. 

In  applying  this  lemma  to  real  minimal  surfaces  we  note  from 

(101)  that 

X     _u  +  v  Y     _u  —  v, 

l-Z=~Y~          l-Z  =     2i 

consequently  the  left-hand  members  of  these  equations  are  equal 
to  ul  and  vl  respectively,  where  u  =  u^  +  ivr  When  the  surface  is 
algebraic  there  exists  an  algebraic  relation  between  the  functions 

X          Y 

, and  each  of  the  Cartesian  coordinates.*    Since,  then, 

7  7 

.L  —  £j       A.  —  /j 

there  is  an  algebraic  relation  between  u^  v^  and  each  of  the 
coordinates  given  by  (110),  it  follows  from  the  lemma  that  each 
of  the  three  expressions 

<k(M)  =  (1  -  ?/)/»  +  2  uf'(u)  -  2/(w), 
fa(u)  =  i (1  +  u2)f"(u)  -  2  iuf'(u)  +  2  if(u), 
4>9(u)  =  2uf"(u)—2f(u) 
are  algebraic  functions  of  w,  and  so  also  isf(u) ;  for, 


Hence  we  have  demonstrated  the  theorem  of  Weierstrass : 

The  necessary  and  sufficient  condition  that  equation  (110)  define  an 
algebraic  surface  is  that  f(u)  be  algebraic. 

*  For.  if  the  surface  is  defined  by  F(x.  y,  z)  =  0,  the  direction-cosines  of  the  normal 

X  Y 

are   functions  of  x,  y,  z.    Eliminating  two  of  the  latter  between      _     >      _  _>  and 

F(x,  y,  z)  =  0,  we  have  a  relation  of  the  kind  described. 


ASSOCIATE  SURFACES  263 

113.  Associate  surfaces.  When  the  equations  of  a  minimal 
surface  S  are  written  in  the  abbreviated  form  (95),  the  linear 
element  is 


This  is  the  linear  element  also  of  a  surface  defined  by 

where  a  is  any  constant.  There  are  an  infinity  of  such  surfaces, 
called  associate  minimal  surfaces.  It  is  readily  found  that  the  direc 
tion-cosines  of  the  normal  to  any  one  have  the  values  (101).  Hence 
any  two  associate  minimal  surfaces  defined  by  (111)  have  their  tan 
gent  planes  at  corresponding  points  parallel,  and  are  applicable. 

Of  particular  interest  is  the  surface  Sl  for  which  a  =  ?r/2.    Its 
equations  are 

i    C 

i)du  —  -    I  (1  —  v2)  <£  (v)  dv, 

(112)          {y  =  -  ^   /  (1  +  u2)  F(u)  du  -  I 
&  J  * 

\  =  i  I  uF(u)  du  —  i  I  v<&  (v)  dv. 

In  order  to  show  that  Sl  is  the  adjoint  (§  109)  of  S,  we  have  only 
to  prove  that  the  asymptotic  lines  on  either  surface  correspond  to 
the  lines  of  curvature  on  the  other.  For  Sl  the  equations  of  the 
lines  of  curvature  and  asymptotic  lines  are 

iF(u)  du2  -  i®  (v)  dv2  =  0, 

respectively.  Comparing  these  with  (103)  and  (104),  we  see  that 
the  desired  condition  is  satisfied. 

From  (98)  and  (112)  we  obtain  the  identities 


\  dx  dxl  +  dy  dyl  +  dz  dz^  =  0. 
The  latter  has  the  following  interpretation : 

On  two  adjoint  minimal  surfaces  at  points  corresponding  with  par 
allelism  of  tangent  planes  the  tangents  to  corresponding  curves  are 
perpendicular. 


264  MINIMAL  SURFACES 

From  (105)  it  follows  that  if  we  put 

u  +  iv  =  /  ^/F(u)  du, 

the  curves  u  =  const,  and  v  =  const,  on  the  surface  are  its  lines  of 
curvature.  Moreover,  for  an  associate  surface  the  lines  of  curva 
ture  are  given  by 

ia  ttf 

R  [e2  (u  +  iv)]  =  const.,  R  [ie  2  (u  -h  iv)]  =  const.t 

or 

—        a      -    .    a  -   .    a      _        a 

u  cos  —  —  v  sin  —  =  const.,  u  sin  —  -f-  v  cos  -  =  const. 
22  22 

From  this  result  follows  the  theorem  : 

The  lines  of  curvature  on  a  minimal  surface  associate  to  a  surface 
S  correspond  to  the  curves  on  S  which  cut  its  lines  of  curvature  under 
the  constant  angle  a/  2. 

Since  equations  (111)  may  be  written 

xa  =  x  cos  a  +  xl  sin  #, 
(114)  -  ^^ycosa  +  ^sina, 

.  za—  z  cos  a  -|-  zl  sin  #, 

the  plane  determined  by  the  origin  of  coordinates,  a  point  P  on  a 
minimal  surface  and  the  corresponding  point  on  its  adjoint,  con 
tains  the  point  Pa  corresponding  to  P  on  every  associate  minimal 
surface.  Moreover,  the  locus  of  these  points  Pa  is  an  ellipse  with  its 
center  at  the  origin.  Combining  this  result  and  the  first  one  of 
this  section,  we  have  *-' 

A  minimal  surface  admits  of  a  continuous  deformation  into  a  series 
of  minimal  surfaces,  and  each  point  of  the  surface  describes  an  ellipse 
whose  plane  passes  through  a  fixed  point  which  is  the  center  of  the 
ellipse. 

114.  Formulas  of  Schwarz.  Since  the  tangent  planes  to  a  minimal 
surface  and  its  adjoint  at  corresponding  points  are  parallel,  we  have 


From  this  and  the  second  of  (113)  we  obtain  the  proportion 

dxl         __         dyl         _         dzl 
Zdy  —  Ydz  =  Xdz  —  Z  dx~  Y  dx  —  X  dy  ' 


FORMULAS  OF  SCHWAKZ 


265 


In  consequence  of  the  first  of  (113)  the  sums  of  the  squares  of  the 
numerators  and  of  the  denominators  are  equal.  And  so  the  com 
mon  ratio  is  -|-1  or  —1.  If  the  expressions  for  the  various  quanti 
ties  be  substituted  from  (98),  (101),  and  (112),  it  is  found  that  the 
value  is  —1.  Hence  we  have 

(115)     dx^Ydz  —  Zdy,    dyl  =  Zdx  —  Xdz,    dz1  =  Xdy  —  Ydx. 
From  these  equations  and  the  formulas  (95),  (112)  we  have 

Zdy  -  Ydz, 


(116) 


and 


(117) 


1  =  x  +  i  £ 

i=y  +  *  (xdz  -  Zdx, 

l  =  z-^-i  \  Ydx  —  Xdy, 

1  =  x—i  \  Zdy  —  Ydz, 


i  \ 


^z  —  i  \Ydx  —  X  dy. 


These  equations  are  known  as  the  formulas  of  Schwarz*  Their 
importance  is  due  to  their  ready  applicability  to  the  solution  of 
the  problem : 

To  determine  a  minimal  surface  passing  through  a  given  curve 
and  admitting  at  each  point  of  the  curve  a  given  tangent  plane.\ 

In  solving  this  problem  we  let  C  be  a  curve  whose  coordinates 
#,  y,  z  are  analytic  functions  of  a  parameter  f,  and  let  JT,  Y,  Z  be 
analytic  functions  of  t  satisfying  the  conditions 

X2  +  F2  +  Z2  =  1,          Xdx  +  Ydy  +  Zdz  =  0. 

*  Crelle,  Vol.  LXXX  (1875),  p.  291. 

t  This  problem  is  a  special  case  of  the  more  general  one  solved  by  Cauchy :  To  deter 
mine  an  integral  surface  of  a  differential  equation  passing  through  a  curve  and  admitting 
at  each  point  of  the  curve  a  given  tangent  plane.  For  minimal  surfaces  the  equation  is 
(84).  Cauchy  showed  that  such  a  surface  exists  in  general,  and  that  it  is  unique  unless  the 
curve  is  a  characteristic  for  the  equation.  His  researches  are  inserted  in  Vols.  XIV,  XV 
of  the  Comptes  Rendus.  The  reader  may  consult  also  Kowalewski,  Theorie  der  partiellen 
Differentialgleichungen,  Crelle,  Vol.  LXXX  (1875),  p.  1;  and  Goursat,  Cours  d' Analyse 
Mathematique,  Vol.  II,  pp.  563-567  (Paris,  1905). 


266  MINIMAL  SUKFACES 

If  xuJ  yu,  zu  denote  the  values  of  x,  y,  z  when  t  is  replaced  by  a 
complex  variable  u,  and  xv,  yv,  zv  the  values  when  t  is  replaced 
by  v,  the  equations 


(118) 

l-  f"(Ydx-Xdy) 

Jv 

define  a  minimal  surface  which  passes  through  C  and  admits  at 
each  point  for  tangent  plane  the  plane  through  the  point  with 
direction-cosines  X,  I7,  Z.  For,  when  u  and  v  are  replaced  by  £, 
these  equations  define  C.  And  the  conditions  (96)  and 


are  satisfied.    Furthermore,  the  surface  defined  by  (118)  affords 
the  unique  solution,  as  is  seen  from  (116)  and  (117). 

When,  in  particular,  C  and  t  are  real,  the  equations  of  the  real 
minimal  surface,  satisfying  the  conditions  of  the  problem,  may  be 
put  in  the  form  ,-  ™  -i 

x  =  R\x  +  il  (Zdy-Ydz}\, 


y  =  R  \y  +  i   C\Xdz  -  Zdx)] , 

z  =  R  \z  +  i  r\Ydx  - Xdy\\ 


As  an  application  of  these  formulas,  we  consider  minimal  surfaces  containing  a 
straight  line.  If  we  take  the  latter  for  the  z-axis,  and  let  0  denote  the  angle  which 
the  normal  to  the  surface  at  a  point  of  the  line  makes  with  the  x-axis,  we  have 

x  =  y  =  0,       z=t,      JT=cos0,       Y=sin<t>,       Z  =  0. 


Hence  the  equations  of  the  surface  are 

x  =  -  RiTsm  <f>dt,        y  =  B{J**C08^ctt,         z  =  R(u). 

Here  <#>  is  an  analytic  function  of  t,  whose  form  determines  the  character  of  the 
surface.  For  two  points  corresponding  to  conjugate  values  of  M,  the  z-coordinates 
are  equal,  and  the  x-  and  ^-coordinates  differ  in  sign.  Hence  : 

Every  straight  line  upon  a  minimal  surface  is  an  axis  of  symmetry. 


FORMULAS  OF  SCHWAKZ  267 

EXAMPLES 

1.  The  tangents  to  corresponding  curves  on  two  associate  minimal  surfaces  meet 
under  constant  angle. 

2.  If  corresponding  directions  on  two  applicable  surfaces  meet  under  constant 
angle,  the  latter  are  associate  minimal  surfaces. 

3.  Show  that  the  catenoid  and  the  right  helicoid  are  adjoint  surfaces  and  deter 
mine  the  function  F(u)  which  defines  the  former. 

4.  Let  C  be  a  geodesic  on  a  minimal  surface  S.    Show  that 
(a)  the  equations  of  the  surface  may  be  put  in  the  form 

y  = 

where  f ,  77,  f  are  the  coordinates  of  a  point  on  C,  and  X,  /*,  v  the  direction-cosines 
of  its  binomial ; 

(6)  if  C'  denotes  the  curve  on  the  adjoint  St  corresponding  to  C,  the  radii  of  first 
and  second  curvature  of  C'  are  the  radii  of  second  and  first  curvature  of  C ; 

(c)  if  C  is  a  plane  curve,  the  surface  is  symmetric  with  respect  to  its  plane. 

5.  The  surface  for  which  F(u)  =  1 is  called  the  surface  of  Henneberg ;  it  is 

u4 
a  double  algebraic  surface  of  the  fifteenth  order  and  fifth  class. 

GENERAL  EXAMPLES 

1.  The  edge  of  regression  of  the  developable  surface  circumscribed  to  two  con- 
focal  quadrics  has  for  projections  on  the  three  principal  planes  the  evolutes  of  the 
focal  conies. 

2.  By  definition  a  tetrahedral  surface  is  one  whose  equations  are  of  the  form 
x  =  A  (u  -  a)m(v  -  a)»,     y  =  B(u-  b)m(v  -  6)n,     z  =  C(u-  c)m(v  -  c)n, 

where  A,  B,  0,  w,  n  are  any  constants.  Show  that  the  parametric  curves  are  con 
jugate,  and  that  the  asymptotic  lines  can  be  found  by  quadratures ;  also  that  when 
m  =  n,  the  equation  of  the  surface  is 

III 
^)>  -  c)  +  (|)"(c  -  a)  +  (0"<a  -  b)  =  (a  -  b)  (b  -  c)  (a  -  c). 

3.  Determine  the  tetrahedral  surfaces,  defined  as  in  Ex.  2,  upon  which  the 
parametric  curves  are  the  lines  of  curvature. 

4.  Find  the  surfaces  normal  to  the  tangents  to  a  family  of  umbilical  geodesies 
on  an  elliptic  paraboloid,  and  find  the  complementary  surface. 

5.  At  every  point  of  a  geodesic  circle  with  center  at  an  umbilical  point  on  the 
ellipsoid  (10)  abc  =  fW<i  (a  +  c  _  r^ 

where  r  is  the  radius  vector  of  the  point  (cf.  §  102). 

6.  The  tangent  plane  to  the  director-cone  of  a  ruled  surface  along  a  generator 
is  parallel  to  the  tangent  plane  to  the  surface  at  the  infinitely  distant  point  on  the 
corresponding  generator. 


268  MINIMAL  SURFACES 

7.  Upon  the  hyperboloid  of  one  sheet,  and  likewise  upon  the  hyperbolic  parab 
oloid,  the  two  lines  of  striction  coincide. 

8.  The  line  of  striction  of  a  ruled  surface  is  an  orthogonal  trajectory  of  the 
generators  only  in  case  the  latter  are  the  binormals  of  a  curve  or  the  surface  is  a 
right  conoid. 

9.  Determine  for  a  geodesic  on  a  developable  surface  the  relation  existing 
between  the  curvature,  torsion,  and  angle  of  inclination  of  the  geodesic  with  the 
generators. 

10.  If  h  denotes  the  shortest  distance  and  a  the  angle  between  two  lines  li  and 
Z2,  and  the  latter  revolves  about  the  former  with  a  helicoidal  motion  of  parameter  a 
(cf .  §  62),  the  locus  of  12  is  a  developable  surface  if  a  =  h  cot  a.    If  a  =  h  tan  a,  the 
surface  is  the  locus  of  the  binormals  of  a  circular  helix. 

11.  If  the  lines  of  curvature  in  one  family  upon  a  ruled  surface  are  such  that 
the  segments  of  the  generators  between  two  curves  of  the  family  are  of  the  same 
length,  the  parameter  of  distribution  is  constant  and  the  line  of  striction  is  a  line 
of  curvature. 

12.  If  two  ruled  surfaces  meet  one  another  in  a  generator,  they  are  tangent  to 
one  another  at  two  points  of  the  generator  or  at  every  point ;  in  the  latter  case  the 
central  point  for  the  common  generator  is  the  same,  and  the  parameter  of  distribu 
tion  has  the  same  value. 

13.  If  tangents  be  drawn  to  a  ruled  surface  at  points  of  the  line  of  striction 
and  in  directions  perpendicular  to  the  generators,  these  tangents  form  the  conju 
gate  ruled  surface.    It  has  the  same  line  of  striction  as  the  given  surface.    More 
over,  a  generator  of  the  given  surface,  the  normal  to  the  surface  at  the  central 
point  C  of  this  generator,  and  the  generator  of  the  conjugate  surface  through  C 
are  parallel  to  the  tangent,  principal  normal,  and  binormal  of  a  twisted  curve. 

14.  Let  C  be  a  curve  on  a  surface  S,  and  S  the  ruled  surface  formed  by  the 
normals  to  S  along  C.    Derive  the  following  results  : 

(a)  the  distance  between  near-by  generators  of  S  is  of  the  first  order  unless  C  is 
a  line  of  curvature ; 

(6)  if  r  denotes  the  distance  from  the  central  point  of  a  generator  to  the  point  of 
intersection  with  S,  rS  (dX)2  —  —  Z  dxd X  ; 

(c)  the  tangent  to  C  at  a  point  M  is  conjugate  to  the  tangent  to  the  surface  at  M 
parallel  to  the  line  of  shortest  distance  ; 

(d)  the  maximum  and  minimum  values  of  r  are  the  principal  radii  of  -S,  pi,  and 
p2,  and  the  above  equation  may  be  written  r  =  pisin2</>  -f  p2  cos2tf>,  where  <f>  is  the 
angle  which  the  corresponding  line  of  shortest  distance  makes  with  the  tangent  to 
the  line  of  curvature  corresponding  to  pz> 

15.  If  C  and  6"  are  two  orthogonal  curves  on  a  surface,  then  at  the  point  of 
intersection  (cf.  Ex.  14)  1111 

rB4>.*'~tf +  *|' 

16.  If  C  and  C'  are  two  conjugate  curves  on  a  surface,  then  at  the  point  of 
intersection  (cf.  Ex.  14)      j      j       i       \  r      R 


GEKEBAL  EXAMPLES  269 

17.  If  two  surfaces  are  applicable,  and  the  radii  of  first  and  second  curvature 
of  every  geodesic  on  one  surface  are  equal  to  the  radii  of  second  and  first  curvature 
of  the  corresponding  geodesic  on  the  other,  the  surfaces  are  minimal. 

1 8.  The  surface  for  which  F  in  (98)  is  constant,  say  3,  is  called  the  minimal  sur 
face  of  Enneper ;  it  possesses  the  following  properties  : 

(a)  it  is  an  algebraic  surface  of  the  ninth  degree  whose  equation  is  unaltered 
when  x,  y,  z  are  replaced  by  y,  x,  —  z  respectively ; 

(6)  it  meets  the  plane  z  =  0  in  two  orthogonal  straight  lines  ; 

(c)  if  we  put  u  =  a  —  i/3,  the  equations  of  the  surface  are 

x  =  3  a  +  3  ap?  -  a3,         y  =  3  ft  +  3  a2  ft  -ft3,        z  =  3  a2  -  3  /32, 

and  the  curves  a  —  const. ,  ft  =  const,  are  the  lines  of  curvature ; 

(d)  the  lines  of  curvature  are  rectifiable  unicursal  curves  of  the  third  order  and 
they  are  plane  curves,  the  equations  of  the  planes  being 

x  +  az  -3a-2a3  =  0,        y  -  ftz  -  3ft  -  2 ^  =  0; 

(e)  the  lines  of  curvature  are  represented  on  the  unit  sphere  by  a  double  family 
of  circles  whose  planes  form  two  pencils  with  perpendicular  axes  which  are  tangent 
to  the  sphere  at  the  same  point ; 

(/)  the  asymptotic  lines  are  twisted  cubics  ; 

(g)  the  sections  of  the  surface  by  the  planes  x  =  0  and  y  =  0  are  cubics,  which 
are  double  curves  on  the  surface  and  the  locus  of  the  double  points  of  the  lines  of 
curvature ; 

(h)  the  associate  minimal  surfaces  are  positions  of  the  original  surface  rotated 
through  the  angle  —  a/2,  about  the  z-axis,  where  a  has  the  same  meaning  as  in  §  113  ; 

(i)  the  surface  is  the  envelope  of  the  plane  normal,  at  the  mid-point,  to  the  join 
of  any  two  points,  one  on  each  of  the  focal  parabolas 

X  =  4 cr,     y  =  0,     z  -  2  a2  -  1 ;         x  -  0,     y  =  4 ft,     z  =  1-2  ft2- 

the  planes  normal  to  the  two  parabolas  at  the  extremities  of  the  join  are  the  planes 
of  the  lines  of  curvature  through  the  point  of  contact  of  the  first  plane. 

19.  Find  the  equations  of  Schwarz  of  a  minimal  surface  when  the  given  curve 
is  an  asymptotic  line. 

20.  Let  S  and  S'  be  two  surfaces,  and  let  the  points  at  which  the  normals  are 
parallel  correspond  ;  for  convenience  let  S  and  S'  be  referred  to  their  common  con 
jugate  system.    Show  that  if  the  correspondence  is  conformal,  either  S  and  S'  are 
homothetic ;  or  both  are  minimal  surfaces ;  or  the  parametric  curves  are  the  lines  of 
curvature  on  both  surfaces,  and  form  an  isothermal  system. 

21.  Find  the  coordinates  of  the  surface  which  corresponds  to  the  ellipsoid  after 
the  manner  of  Ex.  20.    Show  that  the  surface  is  periodic,  and  investigate  the  points 
corresponding  to  the  umbilical  points  on  the  ellipsoid. 

22.  When  the  equations  of  an  ellipsoid  are  in  the  form  (11),  the  curves  u  +  v  = 
const,  lie  on  spheres  whose  centers  coincide  with  the  origin  ;  and  at  all  points  of 
such  a  curve  the  product  pW  is  constant  (§  102). 


CHAPTER  VIII 

SURFACES  OF  CONSTANT  TOTAL  CURVATURE.    W-SURFACES. 
SURFACES  WITH  PLANE  OR   SPHERICAL  LINES  OF  CURVATURE 

115.  Spherical  surfaces  of  revolution.  Surfaces  whose  total  cur 
vature  K  is  the  same  at  all  points  are  called  surfaces  of  constant 
curvature.  When  this  constant  value  is  zero,  the  surface  is  devel 
opable  (§  64).  The  nondevelopable  surfaces  of  this  kind  are  called 
spherical  or  pseudospherical,  according  as  K  is  positive  or  negative. 
We  consider  these  two  kinds  and  begin  our  study  of  them  with 
the  determination  of  surfaces  of  revolution  of  constant  curvature. 

When  upon  a  surface  of  revolution  the  curves  v  =  const,  are 
the  meridians  and  u  =  const,  the  parallels,  the  linear  element  is 
reducible  to  the  form 

(1)  d8*=du*+Gdif, 

where  G  is  a  function  of  u  alone  (§  46).    In  this  case  the  expres 
sion  for  the  total  curvature  (V,  12)  is 

(2)  K  = 

For  spherical  surfaces  we  have  7f=l/a2,  where  a  is  a  real  constant. 
Substituting  this  value  in  equation  (2)  and  integrating,  we  have 


(3) 

where  b  and  c  are  constants  of  integration.  From  (1)  it  is  seen 
that  a  change  in  b  means  simply  a  different  choice  of  the  parallel 
u  =  0.  If  we  take  6  —  0,  the  linear  element  is 

(4)  ds2=du2+  c2cos2-^2. 

a 

From  (III,  99,  100)  it  follows  that  the  equations  of  the  meridian 
curve  are 


-  . 

u  C    it      c2   .  9u 

(5)  r  =  £cos->       z  =  /  \1  —  jsm- 

a  J    \        a2        a 


270 


SPHERICAL  SURFACES  OF  REVOLUTION 


271 


and  that  v  measures  the  angle  between  the  meridian  planes. 
There  are  three  cases  to  be  considered,  according  as  c  is  equal 
to,  greater  than,  or  less  than,  a. 


CASE  I.  c  =  a.    Now 


r  =  a  cos-) 
a 


.    u 

z  —  a  sin  -  > 
a 


FIG.  26 


and  consequently  the  surface  is  a  sphere. 
CASE  II.  c  >  a.  From  the  expression 

for  z  it  follows  that  sin2  -  <  1  and  con- 

a 

sequently  r  >  0.  Hence  the  surface  is 
made  up  of  zones  bounded  by  minimum 
parallels  whose  radii  are  equal  to  the 

?/ 

minimum  value  of  cos  - »  and  the  greatest  parallel  of  each  zone  is 
of  radius  c ;  as  in  fig.  26,  where  the  curves  represent  geodesies. 

CASE  III.  c  <  a.  Now  r  varies  from  0  to  c,  the  former  correspond 
ing  to  the  value  u  =  mcnr/2,  where  m  is  any  odd  integer.  At  these 
points  on  the  axis  the  meridians  meet  the  latter  under  the  angle 

v? 

sin"1-.    Hence  the  surface  is  made  up  of  a  series  of  spindles 
a 

(fig.  27).    For  the  cases  II  and  III  the  expression  for  z  can  be 
integrated  in  terms  of  elliptic  functions.* 

It  is  readily  found  that  these  two  surfaces  are 
applicable  to  the  sphere  with  the  meridians  and 
parallels  of  each  in  correspondence.  Thus,  if  we 
write  the  linear  element  of  the  sphere  in  the  form 

ds2  —  du2  4-  a2  cos2  -  dv'2, 
a 

it  follows  from  (4)  that  the  equations 


u  =  u. 


FIG.  27 


determine  the  correspondence  desired. 
It  is  evident  that  for  values  of  b  other  than  zero  we  should  be 
brought  to  the  same  results.     However,  for  the  sake  of  future 


*Cf.  Bianchi,  Vol.  I,  p.  233. 


272 


SURFACES  OF  CONSTANT  CURVATURE 


reference  we  write  down  the  expressions  for  the  linear  element 
when  b  =  —  7r/2  and  —  Tr/4  together  with  (4),  thus : 


(6) 


(i)   ds2=du* 
(ii)   ds2=du*- 
(iii)   ds2=du* 


cos 


u  TT\  ,  „ 
---  }dv\ 
a  4/ 


Let  S  be  a  surface  with  the  linear  element  (6,  i),  and  consider 
the  zone  between  the  parallels  u0  =  const,  and  rt1  =  const.  A  point 
of  the  zone  is  determined  by  values  of  u  and  v  such  that 


The  parametric  values  of  the  corresponding  point  on  the  sphere 
are  such  that  9  _ 


Hence  when  c  <  «,  the  given  zone  on  S  does  not  cover  the  zone 
on  the  sphere  between  the  parallels  MO  =  const,  and  u^  =  const. ; 
but  when  c  >  a  it  not  only  covers  it,  but  there  is  an  overlapping. 
116.  Pseudospherical  surfaces  of  revolution.  In  order  to  find  the 
pseudospherical  surfaces  of  revolution  we  replace  K  in  (2)  by  —I/a2 
and  integrate.  This  gives 

V5  =  c.  cosh  -  +  <?_  sinh  -  > 
a  a 

where  ct  and  c2  are  constants  of  integration.  We  consider  first 
the  particular  forms  of  the  linear  element  arising  when  either  of 
these  constants  is  zero  or  both  are  equal.  They  may  be  written 


(i)  ds2= 
(ii) 


?/ 
a 

i   oU 

=  du  +  c  sinh2  - 
a 


(iii)  ds^dtf+fe"  dv*. 
Any  case  other  than  these  may  be  obtained  by  taking  for 


either 


of  the  values  cosh   - 


or  sinh(- 


where  b  is  a  constant. 


PSEUDOSPHEKICAL  SURFACES  OF  REVOLUTION    273 

By  a  change  of  the  parameter  u  the  corresponding  linear  elements 
are  reducible  to  (i)  or  (ii).  Hence  the  forms  (7)  are  the  most  general. 
The  corresponding  meridian  curves  are  defined  by 


(8) 


=  c  cosh  -  > 


2  = 


C      .     ,  2  U    , 

sum  -  aw  ; 


w  C    \         <?          .u 

(ii)  r  =  tfsinh-»        2=  I  \  1 -costf-du; 

v  '  a  J    N        a2  a 


(iii)  r  =  cea. 


z  = 


We  consider  these  three  cases  in  detail. 

CASE  I.  The  maximum  and  minimum  values  of  sinh2  -  are  a2/e2 

a      

and  0.  Hence  the  maximum  and  minimum  values  of  r  are  Va2+  c2 
and  c.  At  points  of  a  maximum  parallel  the  tangents  to  the  merid 
ians  are  perpendicular  to  the  axis,  and  at 
points  of  a  minimum  parallel  they  are  par 
allel  to  the  axis.  Hence  the  former  is  a  cus 
pidal  edge,  and  the  latter  a  circle  of  gorge, 
so  that  the  surface  is  made  up  of  spool-like 
sections.  It  is  represented  by  fig.  28,  upon 
which  the  closed  curves  are  geodesic  circles 
and  the  other  curves  are  geodesies.  These 
pseudospherical  surfaces  are  said  to  be  of 
the  hyperbolic  type.* 

CASE  II.  In  order  that  the  surface  be  real 
c'2  cannot  be  greater  than  a2,  a  restriction 
not  necessary  in  either  of  the  other  cases. 
If  we  put  e  =  asino:,f  the  maximum  and 

minimum  values  of  cosh2—  are  cosec2o;  and  1,  and  the  correspond- 

a 

ing  values  of  r  are  a  cos  a  and  0.  The  tangents  to  the  meridians 
at  points  of  the  former  circle  are  perpendicular  to  the  axis,  and  at 
the  points  for  which  r  is  zero  they  meet  the  axis  under  the  angle  a. 
Hence  the  surface  is  made  up  of  a  series  of  parts  similar  in  shape 


FIG.  28 


*  Cf .  Bianchi,  Vol.  I,  p.  223. 


f  Cf.  Bianchi,  Vol.  I,  p.  220. 


274 


SURFACES  OF  CONSTANT  CURVATURE 


to  hour-glasses.  Fig.  29  represents  one  half  of  such  a  part ;  one  of 
the  curves  is  an  asymptotic  line  and  the  others  are  parallel  geodesies. 
The  surface  is  called  a  pseudospherical  surface  of  the  elliptic  type. 

CASE  III.  In  the  preceding  cases  the 
equations  of  the  meridian  curve  can 
be  expressed  without  the  quadrature 
sign  by  means  of  elliptic  functions.* 
In  this  case  the  same  can  be  done  by 
means  of  trigonometric  functions.  For, 


if  we  put 


sin  d)  =  —  ea. 
a 


FIG.  29 


equations  (iii)  of  (8)  become 

(9)  r  =  asin<£,  2  =  a  (log  tan^-f  cos(/>). 


We  find  that  c/>  is  the  angle  which  the  tangent  to  a  meridian  at  a 
point  makes  with  the  axis.  Hence  the  axis  is  an  asymptote  to  the 
curve.  Since  the  length  of  the  segment  of  a  tangent  between  the 
point  of  contact  and  the  intersection  with  the  axis  is  r  cosec  c/> 
or  a,  the  length  of  the  segment  is  independent  of  the  point  of 
contact.  Therefore  the  meridian  curve  is  a  tractrix.  The  surface 
of  revolution  of  a  tractrix  about  its  asymptote  is  called  the  pseudo- 
sphere,  or  the  pseudospherical  surface  of  the 
parabolic  type.  The  surface  is  shown  in 
fig.  30,  which  also  pictures  a  family  of 
parallel  geodesies  and  an  asymptotic  line. 
If  the  integral  (3)  be  written  in  the  form 


u 


=  c,  cos  -  -f-  c«  sin  -  1 
a  a 

the  cases  (i),  (ii),  (iii)  of  (6)  are  seen  to 
correspond  to  the  similar  cases  of  (7).  We 
shall  find  other  marks  of  similarity  between 
these  cases,  but  now  we  desire  to  call  at 
tention  to  differences. 

Each  of  the  three  forms  (7)  determines  a  particular  kind  of 
pseudospherical  surface  of  revolution,  and  c  is  restricted  in  value 


FIG.  30 


*Cf.  Bianchi,  Vol.  I,  pp.  226-228. 


APPLICABILITY  275 

only  for  the  second  case.  On  the  contrary  each  of  the  three  forms 
(6)  serves  to  define  any  of  the  three  types  of  spherical  surfaces  of 
revolution  according  to  the  magnitude  of  c. 

From  (IV,  51)  we  find  that  the  geodesic  curvature  of  the  par 
allels  on  the  surfaces  with  the  linear  elements  (7)  is  measured  by 

the  expressions       -.  -,  -, 

1  .     ,  «*  1      .,  M  1 

-  tann  —  i  -  cotn  — »  -  • 

a  a  a  a  a 

Since  no  two  of  these  expressions  can  be  transformed  into  the 
other  if  u  be  replaced  by  u  plus  any  constant,  it  follows  that  two 
pseudospherical  surfaces  of  revolution  of  different  types  are  not 
applicable  to  one  another  with  meridians  in  correspondence. 

117.  Geodesic  parametric  systems.  Applicability.  Now  we  shall 
show  that  in  corresponding  cases  of  (6)  and  (7)  the  parametric 
geodesic  systems  are  of  the  same  kind,  and  then  we  shall  prove 
that  when  such  a  geodesic  system  is  chosen  for  any  surface 
of  constant  curvature,  not  necessarily  one  of  revolution,  the 
linear  element  can  be  brought  to  the  corresponding  form  of  (6) 
or  (T). 

In  the  first  place  we  recall  that  when  on  any  surface  the  curves 
v  =  const,  are  geodesies,  and  u  =  const,  their  orthogonal  trajectories, 
the  linear  element  is  reducible  to  the  form  (1),  where  G  is,  in 
general,  a  function  of  both  u  and  v ;  and  the  geodesic  curvature 
of  the  curves  u  —  const,  is  given  by  (IV,  51),  namely 

- 

pff 

When,  in  particular,  the  curvature  of  the  surface  is  constant, 

is  given  by  equation  (2)  in  which  K  may  by  replaced  by  ±l/a2. 

Hence,  for  spherical  surfaces,  the  general  form  of  V&  is 

(11)  V&  =  </>  (v)  cos  -  +  A/T  (v)  sin  - , 

a  a 

and  for  pseudospherical  surfaces 

(12)  VG  =  <£  (v)  cosh  -  +  i/r  (v)  sinh  - , 

ci  a 

where  <£  and  i/r  are,  at  most,  functions  of  v.  We  consider  now  the 
three  cases  of  (6)  and  (7). 


276      SURFACES  OF  CONSTANT  CUKVATUKE 

CASE  I.  From  the  forms  (i)  of  (6)  and  (7),  and  from  (10),  it 
follows  that  the  curve  u  =  0  is  a  geodesic  and  that  its  arc  is 
measured  by  cv.  Moreover,  a  necessary  and  sufficient  condition 
that  the  curve  u  =  0  on  any  surface  with  the  linear  element  (1) 
satisfy  these  conditions  is 

=o. 


Applying  these  conditions  to  (11)  and  (12),  we  are  brought  to  the 
forms  (i)  of  (6)  and  (7)  respectively. 

CASE  II.  The  forms  (ii)  of  (6)  and  (7)  satisfy  the  conditions 
=  0, 


0 


which  are  necessary  and  sufficient  that  the  parametric  system  be 
geodesic  polar,  in  which  cv  measures  angles  (cf.  VI,  54).  When 
these  conditions  are  applied  to  (11)  and  (12),  we  obtain  (ii)  of  (6) 
and  of  (7)  respectively. 

CASE  III.  For  (iii)  of  (6)  the  curve  u  =  0  has  constant  geodesic 
curvature  I/a,  and  for  (iii)  of  (7)  all  of  the  curves  u  =  const, 
have  the  same  geodesic  curvature  I/a.  Conversely,  we  find  from 
(11)  and  (12)  that  when  this  condition  is  satisfied  on  any  sur 
face  of  constant  curvature  the  linear  element  is  reducible  to 
one  of  the  forms  (iii).  We  gather  these  results  together  into 
the  theorem : 

The  linear  element  of  any  surface  of  constant  curvature  is  reducible 
to  the  forms  (i),  (ii),  (iii)  of  (6)  or  (7)  according  as  the  parametric 
geodesies  are  orthogonal  to  a  geodesic,  pass  through  a  point,  or  are 
orthogonal  to  a  curve  of  constant  geodesic  curvature. 

When  the  linear  element  of  a  surface  of  constant  curvature  is 
in  one  of  the  forms  (i),  (ii),  (iii)  of  (6)  and  (7),  it  is  said  to  be  of 
the  hyperbolic,  elliptic,  or  parabolic  type  accordingly. 

The  above  theorem  may  be  stated  as  follows : 

Any  spherical  surface  of  curvature  l/az  is  applicable  to  a  sphere 
of  radius  a  in  such  a  way  that  to  a  family  of  great  circles  with 
the  same  diameter  there  correspond  the  geodesies  orthogonal  to  a 


APPLICABILITY  277 

given  geodesic  on  the  surface,  or  all  the  geodesios  through  any 
point  of  it,  or  those  which  are  orthogonal  to  a  curve  of  geodesic 
curvature  I/ a. 

Any  pseudo spherical  surface  of  curvature  —  I/a2  is  applicable  to  a 
pseudospherical  surface  of  revolution  of  any  of  the  three  types ; 
according  as  the  latter  surface  is  of  the  hyperbolic,  elliptic,  or  par 
abolic  type,  to  its  meridians  correspond  on  the  given  surface  geodesies 
which  are  orthogonal  to  a  geodesic,  or  pass  through  a  point,  or  are 
orthogonal  to  a  curve  of  geodesic  curvature  I/a. 

In  the  case  of  spherical  surfaces  one  system  of  geodesies  can 
satisfy  all  three  conditions ;  for  in  the  case  of  the  sphere  the  great 
circles  with  the  same  diameter  are  orthogonal  to  the  equator,  pass 
through  both  poles,  and  are  orthogonal  to  two  small  circles  of 
radius  a/V2,  whose  geodesic  curvature  is  I/a.  But  on  a  pseudo- 
spherical  surface  a  geodesic  system  can  satisfy  only  one  of  these 
conditions.  Otherwise  it  would  be  possible  to  apply  two  surfaces 
of  revolution  of  different  types  in  such  a  way  that  meridians  and 
parallels  correspond. 

From  the  foregoing  theorems  it  follows  that,  in  order  to  carry 
out  the  applicability  of  a  surface  of  constant  curvature  upon  any 
one  of  the  surfaces  of  revolution,  it  is  only  necessary  to  find  the 
geodesies  on  the  given  surface.  The  nature  of  this  problem  is 
set  forth  in  the  theorem : 

The  determination  of  the  geodesic  lines  on  a  surface  of  constant 
curvature  requires  the  solution  of  a  Riccati  equation. 

In  proving  this  theorem  we  consider  first  a  spherical  surface 
defined  in  terms  of  any  parametric  system.  It  is  applicable  to 
a  sphere  of  the  same  curvature  with  center  at  the  origin. 
The  coordinates  of  this  sphere,  expressed  as  functions  of  the 
parameters  u,  v,  can  be  found  by  the  solution  of  a  Riccati  equa 
tion  (§  65).  To  great  circles  on  the  sphere  correspond  geodesic 
lines  on  the  spherical  surface ;  hence  the  finite  equation  of 
the  geodesies  is  ax  +  by  +  cz  =  0,  where  a,  b,  c  are  arbitrary 
constants. 

When  the  surface  is  pseudospherical  we  use  an  imaginary 
sphere  of  the  same  curvature,  and  the  analysis  is  similar. 


278      SURFACES  OF  CONSTANT  CURVATURE 

118.  Transformation  of  Hazzidakis.  Let  a  spherical  surface  of 
curvature  I/a2  be  defined  in  terms  of  isothermal-conjugate  parame 
ters.  Then  * 

D      D"      1 


and  the  Codazzi  equations  (V,  13')  reduce  to 
1  dE      dG      ^dF 

T         _         •/  - — -   II 

dv       dv          du 
From  these  equations  follows  the  theorem : 

The  lines  of  curvature  of  a  spherical  surface  form  an  isothermal- 
conjugate  system. 
For,  a  solution  of  these  equations  is 

E—  G  =  const.,         F—0. 

When  this  constant  is  zero  the  surface  is  a  sphere  because  of  (13). 
Excluding  this  case,  we  replace  the  above  by 

(15)  E  =  a2  cosh2a>,          F  =  0,          G  =  a2  sinh2a>. 

Now 

(16)  D  —  Dn  =  a  sinh  a>  cosh  a>. 

When  these  values  are  substituted  in  the  Gauss  equation  (V,  12), 
namely 


-        _  + 

'  2  #  I  a^  L^  ^     H  ^     HE  du  \    du  ULE  jv     R 

it  is  found  that  o>  must  satisfy  the  equation 

a2ft>    a2o)  A 

/18}  —  H  ---  -  +  smh  a)  cosh  o>  =  0. 

du2      dv2 

Conversely,  for  each  solution  of  this  equation  the  quantities  (15) 
and  (16)  determine  a  spherical  surface. 

If  equations  (14)  be  differentiated  with  respect  to  u  and  v  respec 
tively,  and  the  resulting  equations  be  added,  we  have 


(19)  ^  +  0^  ""  du2     dv2  ' 

*  The  ambiguity  of  sign  may  be  neglected,  as  a  change  of  sign  gives  a  surface  sym 
metrical  with  respect  to  the  origin. 


TRANSFORMATION  OF  HAZZIDAKIS  279 

In  consequence  of  (14)  equation  (17)  is  reducible  to 

4  H 4  (     \\du]      \dv  /  J  L#ti  $v       dv  du 


Equations  (14)  are  unaltered  if  E  and  G  be  interchanged  and  the 
sign  of  F  be  changed.  The  same  is  true  of  (17)  because  of  (19) 
and  (20).  Hence  we  have  : 

If  the  linear  element  of  a  spherical  surface  referred  to  an  isothermal- 
conjugate  system  of  parameters  be 

ds2  =  E  du2+2F  dudv  +  G  dv2, 

there  exists  a  second  spherical  surface  of  the  same  curvature  referred 
to  a  similar  parametric  system  with  the  linear  element 

ds2  =  Gdu2—2F dudv  -f  E dv2, 

and  with  the  same  second  quadratic  form  as  the  given  surface ; 
moreover,  the  lines  of  curvature  correspond  on  the  two  surfaces. 

The  latter  fact  is  evident  from  the  equation  of  the  lines  of  curva 
ture  (IV,  26),  which  reduces  to  Fdu2  +  (G  -  E)  dudv  -  F  dv2  =  0. 
From  (IV,  69)  it  is  seen  that  the  linear  elements  of  the  spherical 
representation  of  the  respective  surfaces  are 

da2  =  -(Gdu2-2F  dudv  +  E  dv2), 

CL 

da2  =  —  (E  du2  +  ZFdudv  -f-  G  dv2). 
a/ 

In  particular  we  have  the  theorem  : 

Each  solution  co  of  equation  (18)  determines  two  spherical  surfaces 
of  curvature  I/ a2;  the  linear  elements  of  the  surfaces  are 
ds2  =  a2  (cosh2  co  du2  +  sinh2  co  dv2), 
ds2  =  a2  (sinh2 co  du2  -f  cosh2 co  dv-), 
and  of  their  spherical  representations 

,9-,  v  {  da2  =  sinh2  co  du2  -f-  cosh2  co  dv\ 

j  d<r*=  cosh2 co  du2  +  sinh2 co  dv2 ; 
moreover,  their  principal  radii  are  respectively 

pl=  a  coth  &),          p2  =  a  tanh  co, 

p[  =  a  tanh  co,         p'2  —  a  coth  co. 


280      SURFACES  OF  CONSTANT  CURVATURE 

Bianchi  *  has  given  the  name  Hazzidakis  transformation  to  the 
relation  between  these  two  surfaces.  It  is  evident  that  the  former 
theorem  defines  this  transformation  in  a  more  general  way. 

119.  Transformation  of  Bianchi.  We  consider  now  a  pseudo- 
spherical  surface  of  curvature  —  I/a2,  defined  in  terms  of  isothermal- 
conjugate  parameters.  We  have 

D__       #'  _  _  1  4 

H~        H~~        a '' 

and  the  Codazzi  equations  reduce  to 

^+  —  -2^=0,         ^+—-2^=0. 

du       du          dv  dv       dv  du 

These  equations  are  satisfied  by  the  values 

(22)  E=  a-  cos2  w,         ^=0,          G  =  a2  sin2  to, 

where  &>  is  a  function  which,  because  of  the  Gauss  equation  (V,  12), 
must  satisfy  the  equation 

o2  <"2 

/c.n,  0  &>        0  (0 

(23)  ___  =  8in»coB«. 

Conversely,  every  solution  of  this  equation  determines  a  pseudo- 
spherical  surface  whose  fundamental  quantities  are  given  by 
(22)  and  by 

(24)  D  =  —  D"  =  —  a  sin  w  cos  to. 

Moreover,  the  linear  element  of  the  spherical  representation  is 

(25)  do-2=sin2o>c^2+cos2a>dv2.  f 

There  is  not  a  transformation  for  pseudospherical  surfaces  sim 
ilar  to  the  Hazzidakis  transformation  of  spherical  surfaces,  but 
there  are  transformations  of  other  kinds  which  are  of  great  im 
portance.  One  of  these  is  involved  in  the  following  theorem  of 
Ribaucour : 

If  in  the  tangent  planes  to  a  pseudo spherical  surface  of  curvature 
—  I/ a'2  circles  of  radius  a  be  described  with  centers  at  the  points  of 
contact,  these  circles  are  the  orthogonal  trajectories  of  an  infinity  of 
surfaces  of  curvature  —  1J a2. 

*  Vol.  II,  p.  437. 

t  This  choice  of  sign  is  made  so  that  the  following  formulas  may  have  the  custom 
ary  form. 


TKANSFOKMATION  OF  BIANCHI  281 

In  proving  this  theorem  we  imagine  the  given  surface  S  referred 
to  its  lines  of  curvature,  and  we  associate  with  it  the  moving  trihe 
dral  whose  axes  are  tangent  to  the  parametric  lines.  From  (22) 
and  (V,  75,  76)  it  follows  that 

rt  n  d(0  3(0 

P  —  0»  P\ =  cos  w»       £  =  sin  »,  £t  =s  0,       r  =  — ,  r1  =  —  » 

cv  vU 

£  =  a  cos  ft),      t]l  =  a  sin  &>,       f  x  =  77  =  0. 

In  the  tangent  z^-plane  we  draw  from  the  origin  M  a  segment 
of  length  #,  and  let  6  denote  its  angle  of  inclination  with  the  #-axis. 
The  coordinates  of  the  other  extremity  M1  with  respect  to  these 
axes  are  a  cos  0,  a  sin  0,  0,  and  the  projections  upon  these  axes  of 
a  displacement  of  Ml  as  M  moves  over  S  are,  by  (V,  51), 

a  —  sin  6  dO  -f  cos  oadu—i  —  du  -\ — -  dv }  sin  6  L 

L  \dv  du      I         J 

a  cos  6  d6  -f  sin  &)  c?y  +( —  rfw  +  —  dv } cos  0  , 
L  \cv  du      /          J 

a  [cos  &>  sin  0  c?v  —  sin  &)  cos  0  du\. 

We  seek  now  the  conditions  which  0  must  satisfy  in  order  that  the 
line  MMl  be  tangent  to  the  locus  of  Ml  denoted  by  S^  and  that  the 
tangent  plane  to  Sl  at  M1  be  perpendicular  to  the  tangent  plane  to 
S  at  M.  Under  these  conditions  the  direction-cosines  of  the  tangent 
plane  to  Sl  with  reference  to  the  moving  trihedral  are 

(26)  sin0,         -COS0,         0, 

and  since  the  tangent  to  the  above  displacement  must  be  in  this 
plane,  we  have 

(27)  dO  +  (—  -  sin  0  cos  co]  du+(—+  cos  0  sin  a>\  dv  =  0. 

\dv  /  \cu  ) 

As   this   equation   must  hold   for  all   displacements   of  Jtf",   it  is 


equivalent  to 

dd      da) 

(  

=  cos  &)  sin  0, 

(28) 

du      dv 

Q/J           /)M 

0"         fft) 

=  —  sin  &)  cos  0. 

dv      du 

These  equations  satisfy  the  condition  of  integrability  in  conse 
quence  of  (23).  Moreover,  0  is  a  solution  of  equation  (23),  as  is 
seen  by  differentiating  equations  (28)  with  respect  to  u  and  v 
respectively  and  subtracting. 


282      SURFACES  OF  CONSTANT  CURVATURE 

By  means  of  (28)  the  above  expressions  for  the  projections  of  a 
displacement  of  M^  can  be  put  in  the  form 

a  cos  0  (cos  ft)  cos  6  du  +  sin  &>  sin  6  dv), 
a  sin  0  (cos  o>  cos  9  du  +  sin  o>  sin  6  dv), 
a  (cos  &)  sin  #  c?t>  —  sin  &)  cos  $  du). 

From  these  it  follows  that  the  linear  element  of  Sl  is 

ds?  =  a2  (cos2  9  du2  +  sin2  0  cfrr2). 

In  order  to  prove  that  Sl  is  a  pseudospherical  surface  referred  to 
its  lines  of  curvature,  it  remains  for  us  to  show  that  the  spherical 
representation  of  these  curves  forms  an  orthogonal  system.  We 
obtain  this  representation  with  the  aid  of  a  trihedral  whose  vertex 
is  fixed,  and  which  rotates  so  that  its  axes  are  always  parallel  to 
the  corresponding  axes  of  the  trihedral  for  S.  The  point  whose 
coordinates  with  reference  to  the  new  trihedral  are  given  by  (26) 
serves  for  the  spherical  representation  of  Sr  The  projections  upon 
these  axes  of  a  displacement  of  this  point  are  reducible,  by  means 
of  (28),  to  cog  e  ^cos  m  sin  0du_  sin  a  cos  0  dv^ 

sin  #(cos  &)  sin  0  du  —  sin  <w  cos  0  dv), 
—  sin  &)  sin  0  du  —  cos  ft)  cos  9  dv, 
from  which  it  follows  that  the  linear  element  is 


Since  0  is  a  solution  of  (23),  the  surface  Sl  is  pseudospherical,  of 
curvature  —  1/«2,  and  the  lines  of  curvature  are  parametric.  To 
each  solution  9  of  equations  (28)  there  corresponds  a  surface  Sr 
Darboux  *  has  called  this  process  of  finding  S1  the  transformation 
of  Bianchi.  As  the  complete  integral  of  equations  (28)  involves  an 
arbitrary  constant,  there  are  an  infinity  of  surfaces  >S\,  as  remarked 
by  Ribaucour.  Moreover,  if  we  put 

(29)  *-tan|. 

these  equations  are  of  the  Riccati  type  in  <£.    Hence,  by  §  14, 

When  one  transform  of  Bianchi  of  a  pseudospherical  surface  is 
known,  the  determination  of  the  others  requires  only  quadratures. 

*  Vol.  Ill,  p.  422. 


TRANSFORMATION  OF  BIANCHI  283 

From  (III,  24)  it  follows  that  the  differential  equation  of  the 
curves  to  which  the  lines  joining  corresponding  points  on  S  and 
$!  are  tangent  is 

(30)  cos  co  smddu  —  sin  o>  cosddv  =  0. 

Hence,  along  such  a  curve,  equation  (27)  reduces  to 

7/1      da)  ,        d(o  i        f. 
d6-\  --  du  H  --  dv  —  0. 
dv  du 

But  from  (VI,  56)  it  is  seen  that  this  is  the  Gauss  equation  of 
geodesies  upon  a  surface  whose  first  fundamental  coefficients 
have  the  values  (22).  Hence  : 

The  curves  on  S  to  which  the  lines  joining  corresponding  points  on 
S  and  Sl  are  tangent  are  geodesies. 

The  orthogonal  trajectories  of  the  curves  (30)  are  defined  by 

(31)  coswcostfdtt  +  sinw'sinflcto  =  0. 

In  consequence  of  (28)  the  left-hand  member  of  this  equation  is  an  exact  differential. 
d£  =  —  a  (cos  w  cos  0du  +  sin  w  sin  6dv), 

the  quantity  e~&a  is  an  integrating  factor  of  the  left-hand  member  of  (30).  Conse 
quently  we  may  define  a  function  rj  thus  : 

drj  =  ae~  £/a  (cos  w  sin  6  du  —  sin  w  cos  6  dv)  . 
In  terms  of  £  and  i\  the  linear  element  of  S  is  expressible  in  the  parabolic  form  (7), 

(32)  <Zs2  =  d£2  +  e^A'cfys. 

Equation  (31)  defines  also  the  orthogonal  trajectories  of  the  curves  on  Si  to 
which  the  lines  MMi  are  tangent,  and  the  equation  of  the  latter  curves  is 


sin  w  cos  6  du  —  cos  w  sin  6  dv  =  0. 
The  quantity  e*/a  is  an  integrating  factor  of  this  equation,  and  if  we  put  accordingly 

d£  =  ae£/a  (sin  w  cos  6  du  —  cos  a;  sin  0  dv)  , 

the  linear  element  of  Si  may  be  expressed  in  the  parabolic  form 
(33)  )  dsf  =  dp  +  e-*£/adp. 

As  the  expressions  (32)  and  (33)  are  of  the  form  of  the  linear  element  of  a  surface  of 
revolution,  the  finite  equations  of  the  geodesies  can  be  found  by  quadratures.   Hence  : 

When  a  Bianchi  transformation  is  known  for  a  surface,  the  finite  equation  of  its 
geodesies  can  be  found  by  quadratures. 

This  follows  also  from  the  preceding  theorem  and  the  last  one  of  §  117. 


•284  SURFACES  OF  CONSTANT  CURVATURE 

120.  Transformation  of  Backlund.  The  transformation  of  Bianchi 
is  only  a  particular  case  of  a  transformation  discovered  by  Backlund,* 
by  means  of  which  from  one  pseudospherical  surface  S  another  S^ 
of  the  same  curvature,  can  be  found.  Moreover,  on  these  two  sur 
faces  the  lines  of  curvature  correspond,  the  join  of  corresponding 
points  is  tangent  at  these  points  to  the  surfaces  and  is  of  constant 
length,  and  the  tangent  planes  at  corresponding  points  meet  under 
constant  angle. 

We  refer  S  to  the  same  moving  trihedral  as  in  the  preceding 
case,  and  let  X  and  6  denote  the  length  of  MMl  and  the  angle 
which  the  latter  makes  with  the  o>axis.  The  coordinates  of  3/x  are 
X  cos  0,  X  sin  0,  0,  and  the  projections  of  a  displacement  of  Ml  are 


(34) 


—  X  sin  0  d0  -f  a  cos  wdu  —  \  sin  6  ( — •  du-\ dv } , 

\0t>  (?M      / 

\cosOdO  -f  a  sin&xi*;  +  X  cos#( —  du  -\ dv ), 

\dv  du     I 


X  (cos  ft)  sin  6  dv  —  sin  o>  cos  6  du)  . 

If  cr  denotes  the  constant  angle  between  the  tangent  planes  tP 
S  and  Sl  at  M  and  Jft  respectively,  since  these  planes  are  to  inter 
sect  in  MMv  the  direction-cosines  of  the  normal  to  Sl  are 

sin  &  sin  0,     —  sin  a  cos  0,     cos  a. 
Hence  0  must  satisfy  the  condition 

X  sin  a-  dB  —  a  sin  or  (cos  G>  sin  6  du  —  sin  &>  cos  0  dv} 


,  7 

-f  X  sin  <r   —  du  H  ---  dv 

\dv  cu 

4-  X  cos  cr  (sin  &)  cos  6du  —  cos  &)  sin  0dv)  =  0. 

Since  this  condition  must  be  satisfied  for  every  displacement,  it  is 
equivalent  to 

X  sin  a  (  --  [-—)  =  #  sin  a-  cos  &)  sin  6  —  X  cos  a  sin  &)  cos  0, 
\dw      fltf/ 

/Q   /I  Q  \ 

X  sin  <r  (  --  h  —  )  =  —  a  sin  <r  sin  w  cos  0  +  X  cos  a  cos  &)  sin  6. 


v      cu 


*Om  ytor  med  konstant  negativ  krokning,  Lunds  Universitets  Arsskrift,  Vol.  XIX 
(1883).  An  English  translation  of  this  memoir  has  been  made  by  Miss  Emily  Coddington 
of  New  York,  and  privately  printed. 


TRANSFORMATION  OF  BACKLUKD  285 

If  these  equations  be  differentiated  with  respect  to  v  and  u  respect 
ively,  and  the  resulting  equations  be  subtracted,  we  have 

a2sin2cr-X2=0, 

from  which  it  follows  that  X  is  a  constant.  Without  loss  of  gen 
erality  we  take  X  =  a  sin  cr.  If  this  value  be  substituted  in  the 
above  equations,  we  have 


(35) 


f   .       (d6      dco\       .    a  a   . 

sin  cr ( =  sin  0  cos  co  —  cos  &  cos  6  sin  o>, 

\du      dv/ 

( 1 )  =  —  cos  0  sin  <w  +  cos  er  sin  0  cos  co, 

\dv      du/ 


smcr 


and  these  equations  satisfy  the  condition  of  integrability.  If  they 
be  differentiated  with  respect  to  u  and  v  respectively,  and  the 
resulting  equations  be  subtracted,  it  is  found  that  0  is  a  solution 
of  (23). 

In  consequence  of  (35)  the  expressions  (34)  reduce  to 

a  cos  0  (cos  &)  cos  0  -f  cos  a  sin  &>  sin  0)  du 

+  a  sin  0  (sin  &>  cos  0  —  cos  a  cos  &>  sin  0)  dv, 
a  cos  0  (cos  co  sin  0  —  cos  cr  sin  &)  cos  0)  C?M 

+  <*•  sin  #(sin  &)  sin  6  +  cos  cr  cos  &)  cos 
a  sin  cr(cosft)  sin0o?v  —  sin  CD  cosOdu), 
and  the  linear  element  of  ^  is 

d**  =  a2  (cos2  <9  dw2  -f-  sin2 


In  a  manner  similar  to  that  of  §  119  it  can  be  shown  that  the 
spherical  representation  of  the  parametric  curves  is  orthogonal, 
and  consequently  these  curves  are  the  lines  of  curvature  on  S^ 

Equations  (35)  are  reducible  to  the  Riccati  form  by  the  change 
of  variable  (29).  Moreover,  the  general  solution  of  these  equations 
involves  two  constants,  namely  cr  and  the  constant  of  integration. 
Hence  we  have  the  theorem  : 

By  the  integration  of  a  Riccati  equation  a  double  infinity  of  pseudo- 
spherical  surfaces  can  be  obtained  from  a  given  surface  of  this  kind. 

We  refer  to  this  as  the  transformation  of  Backlund,  and  indicate 
it  by  Bv,  thus  putting  in  evidence  the  constant  cr. 


286      SURFACES  OF  CONSTANT  CURVATURE 

121.  Theorem  of  permutability.  Let  Sl  be  a  transform  of  S  by 
means  of  the  functions  (0X,  o^).  Since  conversely  S  is  a  transform 
of  Sj,  and  the  equations  for  the  latter  similar  to  (35)  are  reducible 
to  the  Riccati  type,  all  the  transforms  of  Sl  can  be  found  by  quad 
ratures.  But  even  these  quadratures  can  be  dispensed  with  because 
of  the  following  theorem  of  permutability  due  to  Bianehi*: 

If  S1  and  S2  are  transforms  of  S  by  means  of  the  respective  pairs 
of  functions  (01?  a^  and  (02,  <r2),  a  function  <f>  can  be  found  without 
quadratures  which  is  such  that  by  means  of  the  pairs  ((/>,  <r2)  and 
(<£,  o-j)  the  surfaces  Sl  and  S2  respectively  are  transformable  into  a 
pseudospherical  surface  Sf. 

By  hypothesis  </>  is  a  solution  of  the  equations 

sin  o-J  -£-  +  —  *  )=  sin  $  cos  0l  —  cos  cr2  cos  (/>  sin  0^ 

\«H         */ 

/p  I  ^/l  \ 

sin  <T9(  -^-  4-  -^  )  =  —  cos  6  sin  0.  +  cos  cr9  sin  6  cos  ^, 

2 


and  also  of  the  equations 
(37) 


pi  p 

sin  <7,   -—  4-  -— -  =  sn  d>  cos    9—  cos  a.  cos  6  sn 
l*  ' 


=  —  cos  <>  sn      +  coso-Sn)  cos 


The  projections  of  the  line  If^Tf'  on  the  tangents  to  the  lines  of 
curvature  of  Sl  and  on  its  normal,  where  Ml  and  M1  are  correspond 
ing  points  on  $l  and  S',  are 
(38)  a  sin  <r2  cos  </>,          a  sin  <r2  sin  (#>,     ; '   0. 

The  direction-cosines  of  the  tangents  to  the  lines  of  curvature 

of  Sl  with  respect  to  the  line  JOf1?  the  line  MQ^  perpendicular  to 

the  latter  and  in  the  tangent  plane  at  J/,  and  the  normal  to  S  are 

cos  to,          —  cos  <r1  sin  o>,          —  sin  <TI  sin  &), 

sin  a),  cos  cr1  cos  w,  sin  o-j  cos  w. 

From  these  and  (38)  it  follows  that  the  coordinates  of  M'  with 
respect  to  MM^  MQ^  and  the  normal  to   S  are 

a  [sin  a-l  +  sin  cr2  cos  (j>  —  a))],          a  [sin  <r2  cos  o-j  sin  (<f>  —  &))], 

« [sin  o-1  sin  <r2  sin  (</>  —  w)]. 

*  Vol.  II,  p.  418. 


THEOREM  OF  PEEMUTABILITY  287 

Hence  the  coordinates  of  M1  with  respect  to  the  axes  of  the  moving 
trihedral  for  S  are 

x' 

—  =  cos  0j  sin  o-l  4-  cos  0l  sin  &2  cos  (<£  —  &>) 


(39) 


—  sin  0,  sin  <r2  cos  o^  sin  (<£  —  <w), 

=  sin  0l  sin  cr1  +  sin  6l  sin  cr2  cos  ((/>  — 
+  cos  0j  sin  <72  cos  o^  sin  ($  —  <w), 


—  =  sin  cr  sm  &  sin  (9  — 
a 


If  $2  be  transformed  by  means  of  al  and  the  same  function  c£, 
the  coordinates  x",  y",  z"  of  the  resulting  surface  can  be  obtained 
from  (39)  by  interchanging  the  subscripts  1  and  2.  Evidently  z' 
and  z'f  are  equal.  A  necessary  and  sufficient  condition  that  x\  y' 
be  equal  to  #",  y"  respectively  is 

cos  01(d—  x")  +  sin  B^(y'—  y")  —  0, 

cos  02(xr—  x")  +  sin  0z(yr  —  y")  =  0. 

If  the  above  values  be  substituted  in  these  equations,  we  obtain 
[sin  al  cos  (#2  —  0^  —  sin  o-2]  cos  ((/>  —  co) 

—  sin  al  cos  cr2  sin  (#2  —  0^  sin  (^>  —  &>)  =  sin  al  —  sin  cr2  cos  (Qn  —  #J, 

[sin  <72  cos  (^2  —  0j)  —  siu  crj  cos  (</>  —  o>) 

-f-  sin  <72  cos  cr1  sin(^2 —  ^)sin(^)  —  w)  =  sin  crz —  sin  a^  cos(#9 —  6^). 

Solving  these  equations  with  respect  to  sin  (</>  —  «)  and  cos  (</>  —  &>), 
we  get 


sin  o-j  sin  <72  cos  (^2  —  0X)  +  cos  o-j  cos  o-2  —  1 

,.  sin  o-,  sin  o--  +  (cos  <r.  coscr9  —  l)cos(^9  —  ^.) 

cos  (<f>  —  (0)  =  -  -  ^  /i  —  —^  -  -  -  -  -  "  -  r  • 

sin  cr1  sin  <r2  cos  (02  —  c/j)  4-  cos  ^  cos  cr2  —  1 

These  two  expressions  satisfy  the  condition  that  the  sum  of  their 
squares  be  unity,  and  the  function  (j>  satisfies  equations  (36)  and 
(37).  Hence  our  hypotheses  are  consistent  and  the  theorem  of 
permutability  is  demonstrated. 

We  may  replace  the  above  equations  by 


288      SURFACES  OF  CONSTANT  CURVATURE 

The  preceding  result  may  be  expressed  in  the  following  form  : 

When  the  transforms  of  a  given  pseudospherical  surface  are  known, 
all  the  transformations  of  the  former  can  be  effected  by  algebraic 
processes  and  differentiation. 

Thus,  suppose  that  the  complete  integral  of  equations  (35)  is 
(41)  0  =/(w,  v,  <r,  c), 

and  that  a  particular  integral  is 

^i=/(«»  v>  *v  ci)i 

corresponding  to  particular  values  of  the  constants,  and  let  ^ 
denote  the  transform  of  S  by  means  of  ^  and  <7r  All  the  trans 
formations  of  Sl  are  determined  by  the  functions  <f>  and  <r,  where 


Exceptional  cases  arise  when  cr  has  the  value  <rr  For  all  values 
of  c  other  than  cl  formula  (42)  gives  $  =  to  +  WTT,  where  m  is  an 
odd  integer.  When  this  is  substituted  in  equations  (36)  they  re 
duce  to  (35).  In  this  case  S'  coincides  with  S. 

We  consider  now  the  remaining  case  where  c  has  the  value  c1? 
whereupon  the  right-hand  member  of  (42)  is  indeterminate.  In 
order  to  handle  this  case  we  consider  c  in  (41)  to  be  a  function  of 
o-,  reducing  to  cl  for  <r  =  a-^  If  we  apply  the  ordinary  methods  to 

the  function  tan  £L   Ism        ~  *  which  becomes  indeterminate 


for  a  =  o-v  differentiating  numerator  and  denominator  with  respect 
to  <r,  we  have 


or 

/6-w\     .      /a/  .    , 

tan  ^—  —  =  sin          -  4-  c'  — 


.      / 

=  sin  ^ 

V 


where  c'  is  an  arbitrary  constant.    It  is  necessary  to  verify  that  this 
value  of  ^  satisfies  the  equations  (36),  which  is  easily  done.* 


*  Cf  .  Bianchi,  Vol.  II,  p.  418. 


TRANSFORMATION  OF  LIE  289 

122.  Transformation  of  Lie.  Another  transformation  of  pseudo- 
spherical  surfaces  which,  however,  is  analytical  in  character  was 
discovered  by  Lie.*  It  is  immediate  when  the  surface  is  referred 
to  its  asymptotic  lines,  or  to  any  isothermal-conjugate  system 
of  lines. 

Since  the  parameters  in  terms  of  which  the  surface  is  defined  in 
§  119  are  isothermal-conjugate,  the  parameters  of  the  asymptotic 
lines  may  be  given  by 

In  terms  of  these  curvilinear  coordinates  the  linear  elements  of  the 
surface  and  its  spherical  representation  have  the  forms 

ds2  =  a2  (da2  +  2  cos  2  &)  dad/3  +  d/32), 
da-2  =  da2  —  2  cos  2  ft)  dad/3  -f- 
and  equation  (23)  takes  the  form 


dad/3 


—  sin  &)  cos  ft). 


From  the  form  of  this  equation  it  is  evident  that  if  &>  =  <£(#,  ft)  be 
a  solution,  so  also  is  col=  (f>(am,  ft/m)9  where  m  is  any  constant. 
Hence  from  one  pseudospherical  surface  we  can  obtain  an  infinity 
of  others  by  the  transformation  of  Lie.  It  should  be  remarked, 
however,  that  only  the  fundamental  quantities  of  the  new  surfaces 
are  thus  given,  and  that  the  determination  of  the  coordinates  re 
quires  the  solution  of  a  Riccati  equation  which  may  be  different 
from  that  for  the  given  surface. 

Lie  has  called  attention  to  the  fact  that  every  Biicklund  trans 
formation  is  a  combination  of  transformations  of  Lie  and  Bianchi.f 
In  order  to  prove  this  we  effect  the  change  of  parameters  (43)  upon 
equations  (35)  and  obtain 


(44) 


d   ,n         .       1  +  cos  a    .     Q         . 

—  (0  +  co)  =  -  sin  (6  —  &)), 
da  v                     sin  tr 

d    ,a         x      1  —  cos  er    .      Q         x 

—  (6  —  (w)  =  -  :  -  sin  (9  +  <»). 

d3  V  sin  <r 


*Archivfor  Mathematik  og  Naturvidenskab,  Vol.  IV  (1879),  p.  150. 
t  Cf.  Bianchi,  Vol.  II,  p.  434;  Darboux,  Vol.  Ill,  p.  432. 


290       .       SUEFACES  OF  CONSTANT  CUEVATUEE 
In  particular,  for  a  transformation  of  Bianchi  we  have 

—  (B  4-  o))  =  sin  (B  —  o>),         —  (6  —  &>)  =  sin  (6  +  o>). 
ccc  dp 

Suppose  that  we  have  a  pair  of  functions  6  and  &>  satisfying 
these  equations,  and  that  we  effect  upon  them  the  Lie  transforma 
tion  for  which  m  has  the  value  (1  +  cos  cr)/sin  a-.  This  gives 

1  4-  cos  a       1  —  cos  cr 

_ a, — 

sin  or  sin  cr 


/1  + 

\    s 


sin  cr  sm  cr 

As  these  functions  satisfy  (44),  they  determine  a  transformation 
Ba.  But  Ol  may  be  obtained  from  o^  by  effecting  upon  the  latter 
an  inverse  Lie  transformation,  denoted  by  Z"1,  upon  this  result  a 
Bianchi  transformation,  Bn/2,  and  then  a  direct  Lie  transformation, 
Za.  Hence  we  may  write  symbolically 

which  may  be  expressed  thus : 

A  Backlund  transformation  Bff  is  the  transform  of  a   Bianchi 
transformation  ly  means  of  a  Lie  transformation  La* 

EXAMPLES 

1.  The  asymptotic  lines  on  a  pseudospherical  surface  are  curves  of  constant 
torsion. 

2.  Every  surface  whose  asymptotic  lines  are  of  the  same  length  as  their  spherical 
images  is  a  pseudospherical  surface  of  curvature  —  1. 

3.  Show  that  on  the  pseudosphere,  defined  by  (9),  the  curves 

=  0, 


where  6  is  a  constant,  are  geodesies,  and  find  the  radius  of  curvature  of  these  curves. 

4.  When  the  linear  element  of  a  pseudospherical  surface  is  in  the  parabolic 
form  (iii)  of  (7),  the  surface  defined  by 

dx  dy  dz 

x  =  x  —  a  — »        y'  —  y  —  a  — »        z  —  z  —  a  — 

du  cu  du 

is  pseudospherical  (cf .  §  76) ;  it  is  a  Bianchi  transform  of  the  given  surface. 

*  Spherical  surfaces  admit  of  transformations  similar  to  those  of  Lie  and  Backlund. 
The  latter  are  imaginary,  but  such  combinations  of  them  can  be  made  that  the  resulting 
surface  is  real.  For  a  complete  discussion  of  these  the  reader  is  referred  to  chap.  v.  of 
the  Lezioni  of  Bianchi. 


TF-SURFACES  291 


5.  The  helicoids 

X  =  U  COS  V. 


y  =  u  sin u,        z=  f  •*/ Idu  +  hv, 

J    \  a  —  k~u'z      u2 

where  a,  A,  fc  are  constants,  are  spherical  surfaces. 

6.  The  helicoid  whose  meridian  curve  is  the  tractrix  is  called  the  surface  of  Dini. 
Find  its  equations  when  sin  <r  denotes  the  helicoidal  parameter  and  cos  <r  the  con 
stant  length  of  the  segment  of  the  tangent  between  the  curve  and  its  axis.    Show 
that  the  surface  is  pseudospherical. 

7.  The  curves  tangent  to  the  joins  of  corresponding  points  on  a  pseudospherical 
surface  and  on  a  Backlund  transform  are  geodesies  only  when  <r  =  ir/2. 

8.  Let  S  be  a  pseudospherical  surface  and  Si  a  Bianchi  transform  by  means  of 
a  function  d  (§  119).    Show  that 

X{  —  cosw(cos0X1  -f  sin0JT2)  —  sinwJT, 
X'%  =  sin  w  (cos<?  Xi  -f  sin0JT2)  +  coswJT, 
X'  = 


where  .Xi,  X2,  X  are  direction-cosines,  with  respect  to  the  x-axis,  of  the  tangents 
to  the  lines  of  curvature  on  S  and  of  the  normal  to  S,  and  JT{,  X'%,  X'  are  the 
similar  functions  for  Si. 

123.  W-surfaces.  Fundamental  quantities.  Minimal  surfaces 
and  surfaces  of  constant  curvature  possess,  in  common  with  a 
great  many  other  surfaces,  the  property  that  each  of  the  prin 
cipal  radii  is  a  function  of  the  other.  Surfaces  of  this  kind  were 
first  studied  in  detail  by  Weingarten,  *  and,  in  consequence,  are 
called  Weingarten  surfaces,  or  simply  W-surfaces.  Since  the  prin 
cipal  radii  of  surfaces  of  revolution  and  of  the  general  helicoids 
are  functions  of  a  single  parameter  (§§  46,  62),  these  are  TF-surfaces. 
We  shall  find  other  surfaces  of  this  kind,  but  now  we  consider 
the  properties  which  are  common  to  TF-surfaces. 

When  a  surface  S  is  referred  to  its  lines  of  curvature,  the 
Codazzi  equations  may  be  given  the  form 

(45)          glogV^=      1      dp^ 

dv  P2—  Pi  dv  d 

If  a  relation  exists  between  pl  and  />2,  as 


the  integration  of  equations  (45)  is  reducible  to  quadratures,  thus  : 

r  dpi  r  </p2 

=UeJ  *•-*,          V  ^  =  Ve  J  Pl  ~  P2, 

Crelle,  Vol.  LXII  (1863),  pp.  160-173. 


292  JF-SUKFACES 

where  U  and  V  are  functions  of  u  and  v  respectively.  Without 
changing  the  parametric  lines  the  parameters  can  be  so  chosen 
that  the  above  expressions  reduce  to 

r  dp,  f_dpj 

/ 'A  T  \  re       I  a  —  o  r^       I  — 

Thus  £  and  ^  are  expressible  as  functions  of  pl  or  /32,  and  conse 
quently  they  are  functions  of  one  another.  This  relation  becomes 
more  clear  when  we  introduce  an  additional  parameter  K  defined  by 

/•  <*pi 
(48)  *  =  *•'*-* 

By  the  elimination  of  p2  from  this  equation  and  (46)  we  have  a 
relation  of  the  form  —j\(  \ 

When  this  value  is  substituted  in  (48)  we  obtain 


where  the  accent  indicates  differentiation  with  respect  to  K.  From 
(47)  it  follows  that  -,  -. 

V^=±,          ^=T,' 

K  </>' 

When  these  values  are  substituted  in  the  Gauss  equation  for 
the  sphere  (V,  24),  the  latter  becomes 

1/*£'M  ,  jL/*'a«Y    _1.=  0 

du  \  </>"  du)      dv  \tc*  dv)      K<f>' 

This  equation  places  a  restriction  upon  the  forms  of  K  and  <£(«), 
but  it  is  the  only  restriction,  for  the  Codazzi  equations  (45)  are 
satisfied.  Hence  we  have  the  theorem  of  Weingarten  :  * 

When  one  has  an  orthogonal  system  on  the  unit  sphere  for  which 
the  linear  element  is  reducible  to  the  form 


there  exists  a  W-surface  whose  lines  of  curvature  are  represented  by 
this  system  and  whose  principal  radii  are  expressed  by 

(50)  ft  =*(*),        P2  =  *(*)-  «*'(*)• 

•Z.c.,  p.  163. 


FUNDAMENTAL  QUANTITIES  293 

If  the  coordinates  of  the  sphere,  namely  X,  Y,  Z,  are  known 
functions  of  u  and  v,  the  determination  of  the  JF-surface  with  this 
representation  reduces  to  quadratures.  For,  from  the  formulas 
of  Kodrigues  (IV,  32)  we  have 

r     dX  ,  dX  , 

x  =  —     Pi  —  du  +  p2  —  dv, 

J        cu  dv 

C     dY  ,  dY  , 

y  =  —  /  pi  —  du  +  /?2  ^r~  dv, 


r    cz  7         a^  , 

=  ~  /  ft  ~^~  ^  +  P*  v" dv- 

J       du  dv 


The  right-hand  members  of  these  equations  are  exact  differentials, 
since  the  Codazzi  equations  (45)  have  been  satisfied.  If  A",  F, 
Z  are  not  known,  their  determination  requires  the  solution  of  a 
Riccati  equation.  The  relation  between  the  radii  of  the  form  (46) 
is  obtained  by  eliminating  K  from  equations  (50). 

We  find  readily  that  the  fundamental  quantities  for  the  sur 
face  have  the  values 


(51) 


And  from  (48),  (50),  and  (51)  we  obtain 

t  <'pi  _  r  ''Pi 

(52)  Ve  =  p^  "-ft,          vG  =  p,eJ  f'-p>. 

Consider  the  quadratic  form 

(53)  —  [(EJJ'-FD)  du1  +  (El)"-  GD)  dudv  +  (FD"-  GD')  dv*], 
H  . 

which  when  equated  to  zero  defines  the  lines  of  curvature.  When 
these  lines  are  parametric,  this  quadratic  form  is  reducible  by 
means  of  (IV,  74)  to 


But  in  consequence  of  (47)  this  is  further  reducible  for  JF-surfaces  to 
dudv.  Since  the  curvature  of  this  latter  form  is  zero,  the  curvature 
of  (53)  also  is  zero,  and  consequently  (§  135)  the  form  (53)  is  redu 
cible  by  quadratures  to  dudv.  Hence  we  have  the  theorem  of  Lie  : 
The  lines  of  curvature  of  a  W-surface  can  be  found  by  quadratures. 


294  JF-SURFACES 

124.  Evolute  of  a  W-surface.  The  evolute  of  a  JF-surface  pos 
sesses  several  properties  which  are  characteristic.  Referring  to  the 
results  of  §  75,  we  see  that  by  means  of  (52)  the  linear  elements  of 
the  sheets  of  the  evolute  of  a  JF-surface  are  reducible  to  the  form 


or,  in  terms  of  K, 
(55) 


From  these  results  and  the  remarks  of  §  46  we  obtain  at  once 
the  following  theorem  of  Weingarten : 

Each  surface  of  center  of  a  W-surface  is  applicable  to  a  surface  of 
revolution  whose  meridian  curve  is  determined  by  the  relation  between 
the  radii  of  the  given  surface. 

We  have  also  the  converse  theorem,  likewise  due  to  Weingarten  : 

If  a  surface  Sl  be  applicable  to  a  surface  of  revolution,  the  tan 
gents  to  the  geodesies  on  S^  corresponding  to  the  meridians  of  the 
surface  of  revolution  are  normal  to  a  family  of  parallel  W-surfaces; 
if  Sl  be  deformed  in  any  manner  whatever,  the  relation  between  the 
radii  of  these  W-surfaces  is  unaltered. 

In  proving  this  theorem  we  apply  the  results  of  §  76.  If  the 
linear  element  of  Sl  be  rt 

i      i***'  '    ? 

the  principal  radii  of  S  are  given  by 

/tM.  V 

(56)  p^u,         ft-«-^7- 

Since  both  are  functions  of  a  single  parameter,  a  relation  exists 
between  them  which  depends  upon  U  alone,  and  consequently  is 
unaltered  in  the  deformation  of  Sr 

From  (V,  99)  the  projections  upon  the  moving  trihedral  for  8^ 
of  a  displacement  of  a  point  on  the  complementary  surface  £2  are 

U 


£(„___),          0,         (qdu-,   ai~/ir 


EVOLUTE  OF  A  JF-SUBFACE  295 

In  consequence  of  formulas  (V,  48, 75)  the  expression  U(q  du  +  qtdv) 
is  an  exact  differential,  which  will  be  denoted  by  dw.  Hence  the 
linear  element  of  £2  is 
(57)  d»l  =  l 


from  which  it  follows  that  Sz  also  is  applicable  to  a  surface  of 
revolution.* 

The  last  theorem  of  §  75  may  be  stated  thus : 

A  necessary  and  sufficient  condition  that  the  asymptotic  lines  on 
the  surfaces  of  center  S^  S2  of  a  surface  S  correspond  is  that  S  be 
a  W-surface ;  in  this  case  to  every  conjugate  system  on  Sl  or  S2  there 
corresponds  a  conjugate  system  on  the  other. 

From  (V,  98,  98')  it  follows  that  when  S  is  a  TF-surface,  and 
only  in  this  case,  we  have 

(58).         .'.  ^-E^b- 

Hence  at  corresponding  points  the  curvature  is  of  the  same  kind. 

An  exceptional  form  of  equation  (46)  is  afforded  by  the  case  where 
one  or  both  of  the  principal  radii  is  constant.  For  the  plane  both 
radii  are  infinite  ;  for  a  circular  cylinder  one  is  infinite  and  the  other 
has  a  finite  constant  value.  The  sphere  is  the  only  surface  with  both 
radii  finite  and  constant.  For,  if  pr  and  p2  are  different  constants, 
from  (45)  it  follows  that  €  and  ^  are  functions  of  u  and  v  respec 
tively,  which  is  true  only  of  developable  surfaces.  When  one  of  the 
radii  is  infinite,  the  surface  is  developable.  There  remains  the  case 
where  one  has  a  finite  constant  value ;  then  S  is  a  canal  surface  (§  29). 

In  considering  the  last  case  we  take 

then,  from  (48),  we  have 

and  the  linear  element  of  the  sphere  is 

do*  =  ~  +  dv\ 

K 

Conversely,  when  the  linear  element  of  the  sphere  is  reducible  to 
this  form,  the  curves  on  the  sphere  represent  the  lines  of  curvature 
on  an  infinity  of  parallel  canal  surfaces. 

*  Cf.  Darboux,  Vol.  Ill,  p.  329. 


296  TF-SURFACES 

125.  Surfaces  of  constant  mean  curvature.    For  surfaces  of  con 
stant  total  curvature  the  relation  (46)  may  be  written 


where  c  denotes  a  constant.   When  this  value  is  substituted  in  (48) 
we  have,  by  integration, 

(59)  P 

so  that  the  linear  element  of  the  sphere  is 

(60) 


Conversely,  when  we  have  an  orthogonal  system  on  the  sphere  for 
which  the  linear  element  is  reducible  to  the  form  (60),  it  serves  for 
the  representation  of  the  lines  of  curvature  of  a  surface  of  constant 
curvature,  and  of  an  infinity  of  parallel  surfaces. 

When  c  is  positive,  two  of  these  parallel  surfaces  have  constant 
mean  curvature,  as  follows  from  the  theorem  of  Bonnet  (§73).  In 
fact,  the  radii  of  these  surfaces  tifcT 

(61)  pl=^/K*+c±y/~cJ      p9= -=L==  ±V~C. 

v  K  ~r~  c 
If  we  put 

(62)  c  =  a2,  ic  =  a  csch  &>, 

and  replace  u  by  au,  the  linear  element  (60)  becomes 

da--  =  sinlr  co  du2  +  cosh2  o>  dv2.  r« 
In  like  manner,  if  we  replace  u  by  iau,  v  by  iv,  and  take 

(63)  c  =  a2,          K  =  ai  sech  CD, 
the  linear  element  of  the  sphere  is 

da-2  =  cosh2  w  du2  +  sinh2&)  dv*. 

For  the  values  (62)  we  have,  from  (61), 


and  the  linear  elements  of  the  corresponding  surfaces  are 
(65)  <f*a=ffVaw(dwa 


SURFACES  OF  CONSTANT  MEAN  CURVATURE      297 
Moreover,  for  the  values  (63)  the  radii  have  the  values 


cosh  co  sinh  co 

but  the  linear  elements  are  the  same  (65).  In  each  case  the  mean 
curvature  is  ±l/«.  We  state  these  results  in  the  following  form: 

The  lines  of  curvature  upon  a  surface  of  constant  mean  curvature 
form  an  isothermic  system,  the  parameters  of  which  can  be  chosen 
so  that  the  linear  element  has  one  of  the  forms  (65),  where  co  is  a 
solution  of  the  equation 

(67)  l  — ^  4-  — ^  4-  sinh  co  cosh  co  =  0. 

du2      dv2 

Conversely,  each  solution  of  this  equation  determines  two  pair*  of 
applicable  surfaces  of  constant  mean  curvature  ±l/a,  whose  lines 
of  curvature  correspond,  and  for  which  the  radii  p^  p2  of  one  surface 
are  equal  to  the  radii  of  p2,  p^  of  the  applicable  surface. 

It  can  be  shown  that  if  co  =  $(u,  v)  is  a  solution  of  equation  (67), 
so  also  is 

(68)  co1  =  cf)(u  cos  cr  —  v  sin  cr,  u  sin  <r  +  v  cos  cr), 

where  cr  is  any  constant  whatever.  Hence  there  exists  for  spherical 
surfaces  a  transformation  analogous  to  the  Lie  transformation  of 
pseudospherical  surfaces.  This  transformation  can  be  given  a  geo 
metrical  interpretation  if  it  is  considered  in  connection  with  the  sur 
faces  of  constant  mean  curvature  parallel  to  the  spherical  surfaces. 
Let  Sl  denote  the  surface  with  the  linear  element 

(69)  ds2  =  aV  w>  (du2  +  dv2). 
If  we  put 

(70)  uv=u  cos  cr  —  v  sin  cr,          v1=  u  sin  cr  +  v  cos  cr, 
the  solution  (68)  becomes  col  =  cf)(ul,  v^),  and  (69)  reduces  to 

Hence  if  we  make  a  point  (u,  v)  on  S  with  the  linear  element  (65), 
in  which  the  positive  sign  is  taken,  correspond  to  the  point  (uv  vj 
on  8^  the  surfaces  are  applicable,  and  to  the  lines  of  curvature 
u  =  const.,  v  =  const,  on  S  correspond  on  Sl  the  curves 

u  cos  cr  —  v  sin  a  =  const.,          u  sin  cr  -f-  v  cos  cr  =  const. 


298  ^-SURFACES 

But  the  latter  cut  the  lines  of  curvature  u  =  const.,  v  =  const,  on 
Sl  under  the  angle  a-.  Moreover,  the  corresponding  principal  radii 
of  S  and  Sl  are  equal  at  corresponding  points.  Hence  we  have  tha 
following  theorem  of  Bonnet :  * 

A  surface  of  constant  mean  curvature  admits  an  infinity  of  appli 
cable  surfaces  of  the  same  kind  with  preservation  of  the  principal 
radii  at  corresponding  points,  and  the  lines  of  curvature  on  one 
surface  correspond  to  lines  on  the  other  which  cut  the  lires  of 
curvature  under  constant  angle. 

Weingarten  has  considered  the  IF-surfaces  whose  lines  of 
curvature  are  represented  on  the  sphere  by  geodesic  ellipses 
and  hyperbolas.  In  this  case  the  linear  element  of  the  sphere 
is  reducible  to  the  form  (§  90) 

do*  = 

sm*2      C°S*2 

Comparing  this  with  (49),  we  have 

.to  .,  ft) 

/c  =  sin->  <£'=cos- 

'—.  — ' 

from  which  it  follows  that 

to  -f-  sin  ft) 


4 

Hence 

&)  -f  sin  ft)  ft)  —  sin 


and  the  relation  between  the  radii  is  found,  by  the  elimination 
of  w,  to  be 

(72)  2(^-^)=sin2(^+/)2).t 

*  Memoire  sur  la  theorie  des  surfaces  applicables  sur  une  surface  donnce,  Journal  de 
VEcole  Poly  technique,  Cahier  42  (1867) ,  pp.  72  et  seq.  In  this  memoir  Bonnet  solves  com 
pletely  the  problem  of  finding  applicable  surfaces  with  corresponding  principal  radii  equal. 
When  a  surface  possesses  an  infinity  of  applicable  surfaces  of  this  kind,  its  lines  of  curv 
ature  form  an  isothermal  system. 

tDarboux  (Vol.  Ill,  p.  373)  proves  that  these  surfaces  may  be  generated  as  follows: 
Let  C  and  Ci  be  two  curves  of  constant  torsion,  differing  only  in  sign.  The  locus  of  the 
mid-points  M  of  the  join  of  any  points  P  and  PI  of  these  curves  is  a  surface  of  translation. 
If  a  line  be  drawn  through  M  parallel  to  the  intersection  of  the  osculating  planes  of  C  and 
Ci  at  P  and  Pi,  this  line  is  normal  to  a  IP-surface  of  the  above  type  for  all  positions  of  M. 


RULED   JF-SURFACES  299 

126.  Ruled   W-surfaces.    We    conclude    the    present   study    of 
Tr-surfaces  with  the  solution  of  the  problem  : 
To  determine  the  W-surfaces  which  are  ruled. 

This   problem   was    proposed    and    solved    simultaneously    by 
Beltrami*  and  Dini.f    We  follow  the  method  of  the  latter. 

In  §§  106,  107  we  found  that  when  the  linear  element  of  a 
ruled  surface  is  in  the  form 

ds2  =  du2  +  [(u  -  a)2  +  /32]  dv\ 

the  expressions  for  the  total  and  mean  curvatures  are 
/32 

~  = 


where  r  is  a  function  of  v  at  most,  and 

/=(tt~a)a-h^. 

In  order  that  a  relation  exist  between  the  principal  radii  it  is 
necessary  and  sufficient  that  the  equation 

1*  a*     *jr.-l*:-o 

du        dv  dv        du 

be  satisfied  identically.     If  the  above  values  be  substituted,  the 
resulting  equation  reduces  to 
2u  —  a    d  rr2+/3!u-a 


a'l 
\ 


As  this  is  an  identical  equation,  it  is  true  when  u  =  a,  in  which 
case  it  reduces  to  /3'=0.    Hence  /3  is  a  constant  and  the  above 

equation  becomes 

r'  (u  —  of  +  r'ft2  +  /3a"  =  0. 

Since  this  equation  must  be  true  independently  of  the  value  of  w, 
both  r'  and  a"  are  zero.    Therefore  we  have 
(73)  a=cv  +  d,         P  =  e,          r  =  k, 

where  <?,  d,  e,  k  are  constants. 
The  linear  element  is 

ds2  =  du2  +  [(t*  -  cv  -  d)2  +  e2]  dv2. 

*  Annali,  Vol.  VII  (1865),  pp.  13&-150.  t  Annali,  Vol.  VII  (1865),  pp.  205-210. 


300     SURFACES  WITH  PLANE  LINES  OF  CURVATURE 

In  order  to  interpret  this  result  we  calculate  the  expression 
for  the  tangent  of  the  angle  which  the  generators  v  =  const, 
make  with  the  line  of  striction 

u  —  cv  —  d  =  0. 
From  (III,  24)  we  have 

tan  d  =  - ; 

c 

consequently  the  angle  is  constant.  Conversely,  if  6  and  the  param 
eter  of  distribution  j3  be  constant,  a  has  the  form  (73).  Hence  we 
have  the  theorem : 

A  necessary  and  sufficient  condition  that  a  ruled  surface  be  a 
W-surface  is  that  the  parameter  of  distribution  be  constant  and  that 
the  generators  be  inclined  at  a  constant  angle  to  the  line  of  stric 
tion,  which  consequently  is  a  geodesic. 

EXAMPLES 

1.  Show  that  the  helicoids  are  ^surfaces. 

2.  Find  the  form  of  equation  (49),  when  the  surface  is  minimal,  and  show  that 
each  conformal  representation  of  the  sphere  upon  the  plane  determines  a  minimal 
surface. 

3.  Show  that  the  tangents  to  the  curves  v  =  const,  on  a  spherical  surface  with 
the  linear  element  (i)  of  (6)  are  normal  to  a  TT-surface  for  which 

P-2  -  PI  =  COt  -  • 

4.  The   helicoids   are   the  only  >F-surfaces  which  are  such   that  the   curves 
Pi  =  const,  meet  the  lines  of  curvature  under  constant  angle  (cf.  Ex.  23,  p.  188). 

5.  The   asymptotic   lines  on   the  surfaces  of  center  of   a  surface  for  which 
Pl  +  Pz  —  const,  correspond  to  the  minimal  lines  on  the  spherical  representation 
of  the  surface ;  and,  when  />i  —  p2  =  const.,  to  a  rectangular  system  on  the  sphere. 

127.  Spherical  representation  of  surfaces  with  plane  lines  of 
curvature  in  both  systems.  Surfaces  whose  lines  of  curvature  in 
one  or  both  systems  are  plane  curves  have  been  an  object  of  study 
by  many  geometers.  Since  the  tangents  to  a  line  of  curvature  and 
to  its  spherical  representation  at  corresponding  points  are  parallel, 
a  plane  line  of  curvature  is  represented  on  the  sphere  by  a  plane 
curve,  that  is,  a  circle  ;  and  conversely,  a  line  of  curvature  is  plane 
when  its  spherical  representation  is  a  circle. 


SPHERICAL  REPRESENTATION  301 

We  consider  first  the  determination  of  surfaces  with  plane 
lines  of  curvature  in  both  systems  from  the  point  of  view  of 
their  spherical  representation.*  To  this  end  we  must  find  orthog 
onal  systems  of  circles  on  the  sphere.  If  two  circles  cut  one 
another  orthogonally,  the  plane  of  each  must  pass  through  the 
pole  of  the  plane  of  the  other.  Hence  the  planes  of  the  circles 
of  one  system  pass  through  a  point  in  the  plane  of  each  circle 
of  the  second  system,  and  consequently  the  planes  of  each  family 
form  a  pencil,  the  two  axes  being  polar  reciprocal  with  respect  to 
the  sphere.f 

We  consider  separately  the  two  cases :  I,  when  one  axis  is  tan 
gent  to  the  sphere,  and  therefore  the  other  is  tangent  at  the  same 
point  and  perpendicular  to  it ;  II,  when  neither  is  tangent. 

CASE  I.  We  take  the  center  of  the  unit  sphere  for  origin  0,  the 
x-  and  ?/-axes  parallel  to  the  axes  of  the  pencils,  and  let  the  coor 
dinates  of  the  point  of  contact  be  (0,  0,  1).  The  equations  of  the- 
pencils  of  planes  may  be  put  in  the  form 

(74)  x  +  u(z  — 1)=0,          y  +  v(z  — 1)  =  0, 

where  u  and  v  are  the  parameters  of  the  respective  families. 
If  these  equations  be  solved  simultaneously  with  the  equation 
of  the  sphere,  and,  as  usual,  X,  I7,  Z  denote  coordinates  of  the 
latter,  we  have 

v  ^v  r7_u?- 

-  '  ~ 


Now  the  linear  element  of  the  sphere  is 

(T6)  ^='JtXl?- 

CASE  II.  As  in  the  preceding  case,  we  take  for  the  z-axis  the 
common  perpendicular  to  the  axes  of  the  pencils,  and  for  the  x- 
and  ?/-axes  we  take  lines  through  0  parallel  to  the  axes  of  the 
pencils.  The  coordinates  of  the  points  of  meeting  of  the  latter 
with  the  z-axis  are  of  the  form  (0,  0,  a),  (0,  0,  I/a).  The  equa 
tions  of  the  two  pencils  of  planes  could  be  written  in  forms 

*  Bianchi,  Vol.  II,  p.  256;  Darboux,  Vol.  I,  p.  128,  and  Vol.  IV,  p.  180. 
t  Bonnet,  Journal  de  I'Ecole  Poly  technique,  Vol.  XX  (1853),  pp.  136,  137. 


302     SURFACES  WITH  PLANE  LINES  OF  CURVATURE 

similar  to  (74),  but  the  expressions  for  X,  Y,  Z  will  be  found 
to  be  of  a  more  suitable  form  if  the  equations  of  the  families 
of  planes  be  written 

tanw  atanhv 


Proceeding  as  in  Case  I,  we  find 

Vl  —  a2  sin  u 


(77) 


Y=- 
Z  = 


cosh  v  +  a  cos  u 
1  —  a2  sinh  v 


cosh  v  -f  a  cos  u 
cos  u  -\-  a  cosh  v 


cosh  v  -f-  a  cos  w 
and  the  linear  element  is 
(78) 


(cosh  v  +  a  cos  w) 

From  the  preceding  discussion  we  have  tacitly  excluded  the  sys 
tem  of  meridians  and  parallels.  As  before,  the  planes  of  the  two 
families  of  circles  form  pencils,  but  now  the  axis  of  one  pencil 
passes  through  the  center  of  the  sphere  and  the  other  is  at  infinity. 
Hence  this  case  corresponds  to  the  value  zero  for  a  in  Case  II.  In 
fact,  if  we  put  a  =  0  in  (77),  the  resulting  equations  define  a  sphere 
referred  to  a  system  of  meridians  and  parallels,  namely 

_Q  sinw  sinhv  cosw 

(  I  V  )  JL  —  -  -  -  t  JL    —  --  -  -  >  Z/  —  -  -  -  • 

cosh  v  cosh  v  cosh  v 

Since  the  planes  of  the  lines  of  curvature  on  a  surface  are  parallel 
to  the  planes  of  their  spherical  images,  the  curves  v  —  const,  on  a 
surface  with  the  representation  (79)  lie  in  parallel  planes,  and  the 
planes  of  the  curves  u  =  const,  envelop  a  cylinder.  These  surfaces 
are  called  the  molding  surfaces.*  We  shall  consider  them  later. 

128.  Surfaces  with  plane  lines  of  curvature  in  both  systems. 
By  a  suitable  choice  of  coordinate  axes  and  parameters  the 
expressions  for  the  direction-cosines  of  the  normal  to  a  surface 
with  plane  lines  of  curvature  in  both  systems  can  be  given  one 

*  These  surfaces  were  first  studied  by  Monge,  Application  de  L'  Analyse  a  la  Geomt- 
trie,  §  17.  Paris,  1849. 


IN  BOTH  SYSTEMS  303 

of  the  forms  (75)  or  (77).  For  the  complete  determination  of  all 
surfaces  of  this  kind  it  remains  then  for  us  to  find  the  expres 
sion  for  the  other  tangential  coordinate  W,  that  is,  the  distance 
from  the  origin  to  the  tangent  plane.  The  linear  element  of 
the  sphere  in  both  cases  is  of  the  form 

7  2      du2+  dv2 
d(T  =  -  -  -  -> 

where  \  is  such  that 

(80)  -^-  =  0. 

cudv 

From  (VI,  39)  we  see  that  the  equation  satisfied  by  W  is 

gfl      g  log  X  d6  _ 


_  Q 


cucv          dv      du          du      dv 


In  consequence  of  (80),  if  we  change  the  unknown  function  in 
accordance  with  Bl=\0>  the  equation  in  6l  is  of  the  form  (80). 
Hence  the  most  general  value  *for  W  is 


where  U  and  V  are  arbitrary  functions  of  u  and  v  respectively. 

Hence  any  surface  with  plane  lines  of  curvature  in  both  systems 
is  the  envelope  of  a  family  of  planes  whose  equation  is  of  the  form 

(81)  2  ux  +  2  vy  +  (u*+  v2-l)z  =  2  (U+V), 
or 

(82)  Vl  —  a"  sin  ux  —  Vl  —  a2  sinh  vy  +  (cos  u  +  a  cosh  v)  z 

=  (U+  F)Vl-a2. 

The  expressions  for  the  Cartesian  coordinates  of  these  surfaces 
can  be  found  without  quadrature  by  the  methods  of  §  67.  Thus, 
for  the  surface  envelope  of  (81)  we  have  to  solve  for  x,  y,  z  equa 
tion  (81)  and  its  derivatives  with  respect  to  u  and  v.  The  latter  are 

(83)  x  +  uz  =  Z7',         y  +  vz  =  V\ 

where  the  accents  indicate  differentiation.  We  shall  not  carry  out 
this  solution,  but  remark  that  as  each  of  these  equations  contains 
a  single  parameter  they  define  the  planes  of  the  lines  of  curvature. 


304   SURFACES  WITH  PLANE  LINES  OF  CURVATURE 

From  the  form  of  (83)  it  is  seen  that  these  planes  in  each  sys 
tem  envelop  a  cylinder,  and  that  the  axes  of  these  two  cylin 
ders  are  perpendicular.  This  fact  was  remarked  by  Darboux, 
who  also  observed  that  equation  (81)  defines  the  radical  plane 
of  the  two  spheres 

These  are  the  equations  of  two  one-parameter  families  of  spheres, 
whose  centers  lie  on  the  focal  parabolas 

-U,       2/1=0, 


and  whose  radii  are  determined  by  the  arbitrary  functions  U  and  V. 
The  characteristics  of  each  famity  are  defined  by  its  equation  and 
the  corresponding  equation  of  the  pair  (83).  Consequently  the  orig 
inal  surface  is  the  locus  of  the  point  of  intersection  of  the  planes 
of  these  characteristics  and  the  radical  planes  of  the  spheres. 

Similar  results  follow  for  the  equation  (82),  which  defines  the 
radical  planes  of  two  families  of  spheres  whose  centers  are  on  the 
focal  ellipse  and  hyperbola 


(86) 

a;2=0,     2/2  = 

When  in  particular  a  =  0,  these  curves  of  center  are  a  circle  and 
its  axis.  ri 

From  the  foregoing  results  it  follows  that  these  surfaces  may  be 
generated  by  the  following  geometrical  method  due  to  Darboux :  * 

Every  surface  with  plane  lines  of  curvature  in  two  systems  can  be 
obtained  from  two  singly  infinite  families  of  spheres  whose  centers  lie 
on  focal  conies  and  whose  radii  vary  according  to  an  arbitrary  law. 
The  surface  is  the  envelope  of  the  radical  plane  of  two  spheres  S  and  2, 
belonging  to  two  different  families.  If  one  associate  with  S  and  2  two 
infinitely  near  spheres  Sf  and  2',  the  radical  center  of  these  four 
spheres  describes  the  surface  ;  and  the  radical  planes  of  S  and  S'  and 
of  2  and  2'  are  the  planes  of  the  lines  of  curvature. 

*  Vol.  i,  p.  132. 


SURFACES  OF  MONGE  305 

129.  Surfaces  with  plane  lines  of  curvature  in  one  system. 
Surfaces  of  Monge.  When  the  lines  of  curvature  in  one  system 
are  plane,  the  curves  on  the  sphere  are  a  family  of  circles  and 
their  orthogonal  trajectories  ;  and  conversely.  Every  system  of 
this  kind  may  be  obtained  from  a  system  of  circles  and  their 
orthogonal  trajectories  in  a  plane  by  a  stereographic  projection. 
The  determination  of  such  a  system  in  the  plane  reduces  to  the 
integration  of  a  Riccati  equation  (Ex.  11,  p.  50).  Since  the  circles 
are  curves  of  constant  geodesic  curvature  we  have,  in  consequence 
of  the  first  theorem  of  §  84,  the  theorem  : 

The  determination  of  all  the  surfaces  with  plane  lines  of  curva 
ture  in  one  system  requires  the  solution  of  a  Riccati  equation  and 
quadratures. 

We  shall  discuss  at  length  several  kinds  of  surfaces  with  plane 
lines  of  curvature  in  one  system,  and  begin  with  the  case  where 
these  curves  are  geodesies.  They  are  consequently  normal  sections 
of  the  surface.  Their  planes  envelop  a  developable  surface,  called 
the  director-developable,  and  the  lines  of  curvature  in  the  other  sys 
tem  are  the  orthogonal  trajectories  of  these  planes.  Conversely, 
the  locus  of  any  simple  infinity  of  the  orthogonal  trajectories  of  a 
one-parameter  system  of  planes  is  a  surface  of  the  kind  sought. 
For,  the  planes  cut  the  surface  orthogonally,  and  consequently 
they  are  lines  of  curvature  and  geodesies  (§  59).  Since  these 
planes  are  the  osculating  planes  of  the  edge  of  regression  of 
the  developable,  the  orthogonal  trajectories  can  be  found  by 
quadratures  (§  17). 

Suppose  that  we  have  such  a  surface,  and  that  C  denotes  one  of 
the  orthogonal  trajectories  of  the  family  of  plane  lines  of  curvature. 
Let  the  coordinates  of  C  be  expressed  in  terms  of  the  arc  of  the 
curve  from  a  point  of  it,  which  will  be  denoted  by  v  .  As  the 
plane  of  each  plane  line  of  curvature  F  is  normal  to  C  at  its  point 
of  meeting  with  the  latter,  the  coordinates  of  a  point  P  of  F  with 
reference  to  the  moving  trihedral  of  C  are  0,  77,  f.  Since  P  describes 
an  orthogonal  trajectory  of  the  planes,  we  must  have  (I,  82) 


dv 


306      SURFACES  WITH  PLANE  LINES  OF  CURVATURE 

where  r  denotes  the  radius  of  torsion  of  C.    If  we  change  the 
parameter  of  C  in  accordance  with  the  equation 


the  above  equations  become 


The  general  integral  of  these  equations  is 

(88)          ?;  =  U^  cos  vl  —  U2  sin  v^       f  =  U^  sin  vl  -f  ?72  cos  vlt 

where  C^  and  ?72  are  functions  of  the  parameter  u  of  points  of  F. 
When  v  =  0  we  have  vl  =  0,  and  so  the  curve  F  in  the  plane  through 
the  point  v  =  0  of  C  has  the  equations  77  =  U^  ?=  Z72.  Hence  the 
character  of  the  functions  U^  and  U2  is  determined  by  the  form  of 
the  curve ;  and  conversely,  the  functions  U}  and  U2  determine  the 
character  of  the  curve. 

By  definition  (87)  the  function  vt  measures  the  angle  swept 
out  in  the  plane  normal  to  C  by  the  binormal  of  the  latter,  as  this 
plane  moves  from  v  =  0  to  any  other  point.  Hence  equations  (88) 
define  the  same  curve,  in  this  moving  plane,  for  each  value  of  v^ 
but  it  is  defined  with  respect  to  axes  which  have  rotated  through 
the  angle  vr  Hence  we  have  the  theorem  : 

Any  surface  whose  lines  of  curvature  in  one  system  are  geodesies 
can  be  generated  by  a  plane  curve  whose  plane  rolls,  without  slipping, 
over  a  developable  surface. 

These  surfaces  are  called  the  surfaces  of  Monge,  by  whom  they 
were  first  studied.  He  proposed  the  problem  of  finding  a  surface 
with  one  sheet  of  the  e volute  a  developable.  It  is  evident  that  the 
above  surfaces  satisfy  this  condition.  Moreover,  they  furnish  the 
only  solution.  For,  the  tangents  to  a  developable  along  an  ele 
ment  lie  in  the  plane  tangent  along  this  element,  and  if  these 
tangents  are  normals  to  a  surface,  the  latter  is  cut  normally  by 
this  plane,  and  consequently  the  curve  of  intersection  is  a  line  of 
curvature.  In  particular,  a  molding  surface  (§  127)  is  a  surface 
of  Monge  with  a  cylindrical  director-developable. 

Since  every  curve  in  the  moving  plane  of  the  lines  of  curva 
ture  generates  a  surface  of  Monge,  a  straight  line  in  this  plane 


MOLDING  SURFACES  307 

generates  a  developable  surface  of  Monge.  For,  all  the  normals 
to  the  surface  along  a  generator  lie  in  a  plane  (§  25).  Hence: 

A  necessary  and  sufficient  condition  that  a  curve  F  in  a  plane 
normal  to  a  curve  C  at  a  point  Q  generate  a  surface  of  Monge  as 
the  plane  moves,  remaining  normal  to  the  curve,  is  that  the .  line 
joining  a  point  of  T  to  Q  generate  a  developable. 

130.  Molding  surfaces.  When  the  orthogonal  trajectory  C  is  a 
plane  curve,  the  planes  of  the  curves  F  are  perpendicular  to  the 
plane  of  C,  and  consequently  the  director-developable  is  a  cylinder 
whose  right  section  is  the  plane  evolute  of  C.  The  surface  is  a 
molding  surface  (§  127),  and  all  the  lines  of  curvature  of  the  sec 
ond  system  are  plane  curves,  —  involutes  of  the  right  section  of 
the  cylinder.  Hence  a  molding  surface  may  be  generated  by  a 
plane  curve  whose  plane  rolls  without  slipping  over  a  cylinder. 
We  shall  apply  the  preceding  formulas  to  this  particular  case. 

Since  1/r  is  equal  to  zero,  it  follows  from  (88)  that  ?;  and  £  are 
functions  of  u  alone.  If  u  be  taken  as  a  measure  of  the  arc  of  the 
curve  F,  we  have,  in  all  generality, 

?;  =  U,          f  =  I  Vl  —  U''2  du, 

where  the  function  U  determines  the  form  of  F.  If  we  take  the 
plane  of  the  curve  C  for  2  =  0,  and  XQ,  yQ  denote  the  coordinates 
of  a  point  of  C,  the  equations  of  the  surface  may  be  written 

x  =  x0  +  U  cos  v,         y  =  2/o  +  u  sin  vi          2  =  /  Vl  —  U''2  du, 

where  v  denotes  the  angle  which  the  principal  normal  to  C  makes  with 

the  a>axis.    Since     ^x  ^(, 

—^  =  sin  v         — —  =  —  cos  v, 

if  V  denote  the  radius  of  curvature  of  C,  then  dsQ  =  V  dv,  and  the 
equations  of  the  surface  can  be  put  in  the  following  form,  given  by 

Darboux  * :  (  '  r 

v  -f-  I   Fsin  v  dv, 

J 

(89) 


=  U  sin  v  —  I  V  cos  v  dv, 


*  Vol.  I,  p.  105. 


308      SURFACES  WITH  PLANE  LINES  OF  CURVATURE 
The  equations  of  the  right  section  of  the  cylinder  are 
x  =  XQ  +  V  cos  v  =  I  V  cos  v  dv, 

y  =  yQ  -f-  V  sin  v  =  I  V'  sin  v  dv. 

In  passing,  we  remark  that  surfaces  of  revolution  are  molding  sur 
faces,  whose  director-cylinder  is  a  line  ;  this  corresponds  to  the 
case  V  —  0. 

EXAMPLES 

1.  When  the  spherical  representation  of  the  lines  of  curvature  of  a  surface  is 
isothermal  and  the  curves  in  one  family  on  the  sphere  are  circles,  the  curves  in  the 
other  family  also  are  circles. 

2.  If  the  lines  of  curvature  in  one  system  on  a  minimal  surface  are  plane,  those 
in  the  other  system  also  are  plane. 

3.  Show  that  the  surface 

x  —  au  _|_  sin  u  cosh  v,     y  =  v  +  a  cos  u  sinh  v,     z  —  V 1  —  a'2  cos  u  cosh  v, 
is  minimal  and  that  its  lines  of  curvature  are  plane.    Find  the  spherical  representa 
tion  of  these  curves  and  determine  the  form  of  the  curves. 

4.  Show  that  the  surface  of  Ex.  3  and  the  Enneper  surface  (Ex.  18,  p.  209)  are 
the  only  minimal  surfaces  with  plane  lines  of  curvature. 

5.  When  the  lines  of  curvature  in  one  system  lie  in  parallel  planes,  the  surface 
is  of  the  molding  type. 

6.  A  necessary  and  sufficient  condition  that  the  lines  of  curvature  in  one  system 
on  a  surface  be  represented  on  the  unit  sphere  by  great  circles  is  that  it  be  a  sur 
face  of  Monge. 

7.  Derive  the  expressions  for  the  point  coordinates  of  a  molding  surface  by  the 
method  of  §  67. 

131.  Surfaces  of  Joachimsthal.  Another  interesting  class  of 
surfaces  with  plane  lines  of  curvature  in  one  system  are  those  for 
which  all  the  planes  pass  through  a  straight  line.  Let  one  of  these 
lines  of  curvature  be  denoted  by  F,  and  one  of  the  other  system 
by  C.  The  developable  enveloping  the  surface  along  the  latter  has 
for  its  elements  the  tangents  to  the  curves  F  at  their  points  of 
intersection  with  0.  Since  these  elements  lie  in  the  planes  of  the 
curves  F,  the  developable  is  a  cone  with  its  vertex  on  the  line  Z>, 
through  which  all  these  planes  pass.  This  cone  is  tangent  to  the 
surface  along  (7,  and  its  elements  are  orthogonal  to  the  latter.  Con 
sequently  C  is  the  intersection  of  the  surface  and  a  sphere  with 


SURFACES  OF  JOACHIMSTHAL  309 

center  at  the  vertex  of  the  cone  which  cuts  the  surface  orthogo 
nally.  Hence  we  have  the  following  result,  due  to  Joachimsthal  * : 

When  the  lines  of  curvature  in  one  system  lie  in  planes  passing 
through  a  line  D,  the  lines  of  curvature  in  the  second  system  lie  on 
spheres  whose  centers  are  on  D  and  which  cut  the  surface  orthogonally. 

Such  surfaces  are  called  surfaces  of  Joachimsthal.  Each  of  the 
curves  of  the  first  system  is  an  orthogonal  trajectory  of  the  circles 
in  which  the  spheres  are  cut  by  its  plane.  Therefore,  in  order  to 
derive  the  equations  of  such  a  surface,  we  consider  first  the  orthog 
onal  trajectories  of  a  family  of  circles  whose  centers  are  on  a  line. 
If  the  latter  be  taken  for  the  ?;-axis,  the  circles  are  defined  by 

f  =  r  sin  0,         77  =  r  cos  6  +  u, 

where  r  denotes  the  radius,  6  the  angle  which  the  latter  makes 
with  the  ?;-axis,  and  u  the  distance  of  the  center  from  the  origin. 
Now  r  is  a  function  of  u,  and  6  is  independent  of  u.  In  order  that 
these  same  equations  may  define  an  orthogonal  trajectory  of  the 
circles,  6  must  be  such  a  function  of  u  that 

cos  0^- sin  0^  =  0, 

du  cu 

or 

rf^_sin0  =  0. 
du 

By  integration  we  have 

(90)  tan|  =  F</r, 

where  V  denotes  the  constant  of  integration. 

Since  each  section  of  a  surface  of  Joachimsthal  by  a  plane 
through  its  axis  is  an  orthogonal  trajectory  of  a  family  of  circles 
whose  centers  are  on  this  axis,  the  equations  of  the  most  general 
surface  of  this  kind  are  of  the  form 

x  =  r  sin  6  cos  v,       y  =  r  sin  6  sin  v,       z  —  u  -f  r  cos  #, 

where  v  denotes  the  angle  which  the  plane  through  a  point  and 
the  axis  makes  with  the  plane  y  —  0,  and  6  is  given  by  (90),  in 
which  now  V  is  a  function  of  v. 

*  Crelle,  Vol.  LIV  (1857),  pp.  181-192. 


310      SURFACES  WITH  PLANE  LINES  OF  CURVATURE 

When  V  is  constant  0  is  a  function  of  u  alone,  and  the  surface 
is  one  of  revolution.  For  other  forms  of  Vihe  geometrical  genera 
tion  of  the  surfaces  is  given  by  the  theorem  : 

Given  the  orthogonal  trajectories  of  a  family  of  circles  whose  cen 
ters  lie  on  a  right  line  D  ;  if  they  be  rotated  about  D  through  dif 
ferent  angles,  according  to  a  given  law,  the  locus  of  the  curves  is  a 
surface  of  Joachimsthal. 

132.  Surfaces  with  circular  lines  of  curvature.  We  consider 
next  surfaces  whose  lines  of  curvature  in  one  system  are  circles. 
Let  o-  denote  the  constant  angle  between  the  plane  of  the  circle 
C  and  the  tangent  planes  to  the  surface  along  C  (cf.  §  59),  p  the 
radius  of  normal  curvature  in  the  direction  of  C,  and  r  the  radius 
of  the  latter.  Now  equation  (IV,  17)  may  be  written 

(91)  r  =  p  sin  a. 

As  an  immediate  consequence  we  have  the  theorem  : 

A  necessary  and  sufficient  condition  that  a  plane  line  of  curvature 
be  a  circle  is  that  the  normal  curvature  of  the  surface  in  its  direction 
be  the  same  at  all  of  its  points. 

Since  the  normals  to  the  surface  along  C  are  inclined  to  its  plane 
under  constant  angle,  they  form  a  right  circular  cone  whose  vertex 
is  on  the  axis  of  C.  Moreover,  the  cone  cuts  the  surface  at  right 
angles,  and  consequently  the  sphere  of  radius  p  and  center  at  the 
vertex  of  the  cone  is  tangent  to  the  surface  along  C.  Hence  the 
surface  is  the  envelope  of  a  family  of  spheres  pf  variable  or  con 
stant  radius,  whose  centers  lie  on  a  curve. 

Conversely,  we  have  seen  in  §  29  that  the  characteristics  of 
the  family  of  spheres 


where  x,  y,  z  are  the  coordinates  of  a  curve  expressed  in  terms  of  its 
arc,  and  11  is  a  function  of  the  same  parameter,  are  circles  of  radius 


(92)  r 

whose  axes  are  tangent  to  the  curve  of  centers  and  whose  centers 
have  the  coordinates 

(93)  xl  =  x  -  aRR',      y 


CIRCULAR  LIKES  OF  CURVATURE  311 

where  a,  ft,  7  are  the  direction-cosines  of  the  axis,  and  the  accent 
indicates  differentiation.  The  normals  to  the  envelope  along  a 
characteristic  form  a  cone,  and  consequently  these  circles  are  lines 
of  curvature  upon  it.  Hence  : 

A  necessary  and  sufficient  condition  that  the  lines  of  curvature  in 
one  family  be  circles  is  that  the  surface  be  the  envelope  of  a  single 
infinity  of  spheres,  the  locus  of  whose  centers  is  a  curve,  the  radii 
being  determined  by  an  arbitrary  law. 

From  equations  (91),  (92)  it  follows  that  R'  —  cos  a.  Hence  the 
circles  are  geodesies  only  when  R  is  constant,  that  is,  for  canal 
surfaces  (§  29).  In  this  case,  as  is  seen  from  (92),  all  the  circles 
are  equal. 

The  circles  are  likewise  of  equal  radius  a  when 


where  s  is  the  arc  of  the  curve  of  centers  and  c  is  a  constant  of 
integration.    Now  equations  (93)  become 

^  =  x  —  (s  +  c)  a.         yl=^y  —  (s  +  c}IB,          zl=z  —  (s  +  c)y, 

which  are  the  equations  also  of  an  involute  of  the  curve  of  centers 
(§  21).    This  result  may  be  stated  thus*  : 

If  a  string  be  unwound  from  a  curve  in  such  a  way  that  its  moving 
extremity  M  generates  an  involute  of  the  curve,  and  if  at  M  a  circle 
be  constructed  whose  center  is  M  and  whose  plane  is  normal  to  the 
string,  then  as  the  string  is  unwound  this  circle  generates  a  surface 
with  a  family  of  equal  circles  for  lines  of  curvature. 

The  locus  of  the  centers  of  the  spheres  enveloped  by  a  surface  is 
evidently  one  sheet  of  the  evolute  of  the  surface,  and  the  radius 
of  the  sphere  is  the  radius  of  normal  curvature  in  the  direction 
of  the  circle.  Consequently  this  radius  is  a  function  of  the 
parameter  of  the  spheres.  Conversely,  from  §  75,  we  have  that 
when  £2  is  a  curve  H2  =  0,  and  consequently 


Cf.  Bianchi,  Vol.  II,  p.  272. 


312      SURFACES  WITH  PLANE  LINES  OF  CURVATURE 

Excluding  the  case  of  the  sphere,  we  have  that  p.2  is  a  function  of 
u  alone.    From  the  formulas  of  Rodrigues  (IV,  32), 

2x  _          dX          dy  _          dY          dz  _          a£ 

d^~~~Pz~^1      fo~~p2~fo'      Tv~~pz^v' 

we  have,  by  integration, 


Hence  the  points  of  the  surface  lie  on  the  spheres 
(x  -  U,  )a  +  (y  -  tg2  +  (z  -  P,)'  =  ft«, 
and  the  spheres  are  tangent  to  the  surface. 

Since  the  normals  to  a  surface  along  a  circular  line  of  curvature 
form  a  cone  of  revolution,  the  second  sheet  of  the  e  volute  is  the 
envelope  of  a  family  of  such  cones.  The  characteristics  of  such  a 
family  are  conies.  Hence  we  have  the  theorem  : 

A  necessary  and  sufficient  condition  that  one  sheet  of  the  evolute  of 
a  surface  be  a  curve  is  that  the  surface  be  the  envelope  of  a  single  infinity 
of  spheres  ;  the  second  focal  sheet  is  the  locus  of  a  family  of  conies. 

133.  Cyclides  of  Dupin.  From  the  preceding  theorem  it  results 
that  if  also  the  second  sheet  of  the  evolute  of  a  surface  be  a  curve, 
it  is  a  conic,  and  then  the  first  sheet  also  is  a  conic.  Moreover,  these 
conies  are  so  placed  that  the  cone  formed  by  joining  any  point  on 
one  conic  to  all  the  points  of  the  other  is  a  cone  of  revolution. 
A  pair  of  focal  conies  is  characterized  by  this  property.  And  so 
we  have  the  theorem  : 

A  necessary  and  sufficient  condition  that  the  lines  of  curvature  in 
both  families  be  circles  is  that  the  sheets  of  the  evolute  be  a  pair  of 
focal  conies.* 

These  surfaces  are  called  the  cy  elides  of  Dupin.    They  are  the 
envelopes  of  two  one-parameter  families  of  spheres,  and  all  such 
envelopes  are  cyclides  of  Dupin.    A  sphere  of  one  family  touches 
each  sphere  of  the  other  family.    Consequently  the  spheres  of  which 
the  cyclide  is  the  envelope  are  tangent  to  three  spheres. 
We  shall  prove  the  converse  theorem  of  Dupin  f  : 
The  envelope  of  a  family  of  spheres  tangent  to  three  fixed  spheres 
is  a  cyclide. 

*  Cf.  Ex.  19,  p.  188. 

t  Applications  de  geomttrie  et  de  mechanique,  pp.  200-210.    Paris,  1822. 


CYCLIDES  OF  DUPIN  313 

The  plane  determined  by  the  centers  of  the  three  spheres  cuts 
the  latter  in  three  circles.  If  any  point  on  the  circumference  (7, 
orthogonal  to  these  circles,  be  taken  for  the  pole  of  a  transforma 
tion  by  reciprocal  radii  (cf.  §  80),  C  is  transformed  into  a  straight 
line  L.  Since  angles  are  preserved  in  this  transformation,  the  three 
fixed  spheres  are  changed  into  three  spheres  whose  centers  are  on  L. 
Evidently  the  envelope  of  a  family  of  spheres  tangent  to  these  three 
spheres  is  a  tore  with  L  as  axis.  Hence  the  given  envelope  is  trans 
formed  into  a  tore.  However,  the  latter  surface  is  the  envelope  of 
a  second  family  of  spheres  whose  centers  lie  on  L.  Therefore,  if 
the  above  transformation  be  reversed,  we  have  a  second  family  of 
spheres  tangent  to  the  envelope,  and  so  the  latter  is  a  cyclide  of 
Dupin.  We  shall  now  find  the  equations  of  these  surfaces. 

Let  (x^  y^  zj  and  (#2,  y2,  z2)  denote  the  coordinates  of  the  points 
on  the  focal  conies  which  are  the  curves  of  centers  of  the  spheres, 
and  jR1?  E2  the  radii  of  the  spheres.  The  condition  of  tangency  is 
(94)  (Xi-x 


We  consider  first  the  case  where  the  evolute  curves  are  the  focal 
parabolas  defined  by  (85).    Now  equation  (94)  reduces  to 


Since  2il  and  Rz  are  functions  of  u  and  v  respectively,  this  equation 
is  equivalent  to 


where  a  is  an  arbitrary  constant  whose  variation  gives  parallel 
surfaces. 

By  the  method  of  §  132  we  find  that  the  coordinates  (f,  77,  f)  of 
the  centers  of  the  circular  lines  of  curvature  u  —  const,  and  the 
radius  p  are 


9-0, 


314    SURFACES  WITH  PLANE  LINES  OF  CURVATURE 

Hence  if  P  be  a  point  on  the  circle  and  6  denote  the  angle  which 
the  radius  to  P  makes  with  the  positive  direction  of  the  normal  to 
the  parabola  (85),  the  coordinates  of  P  are 

x  =  f  H  —  ^         cos  0,       ^  =  p  sin  0,       2  =  ?  --  cos  9. 

v  1  +  u2  Vl  +  if 

This  surface  is  algebraic  and  of  the  third  order. 

If  the  evolute  curves  are  the  focal  ellipse  and  hyperbola  (86),  we  have 

(96)  Rl  =  -  (a  cos  u  +  *),          A'2  =  -  (cosh  v  —  /c), 

-!  - 

where  /c  is  an  arbitrary  constant  whose  variation  gives  parallel  surfaces. 
This  cyclide  of  Dupin  is  of  the  fourth  degree.  When  in  particular 
the  constanta  is  zero,  the  surface  is  the  ordinary  tore,  or  anchor  ring.* 

134.  Surfaces  with  spherical  lines  of  curvature  in  one  system. 
Surfaces  with  circular  lines  of  curvature  in  one  system  belong  evi 
dently  to  the  general  class  of  surfaces  with  spherical  lines  of  curva 
ture  in  one  system.  We  consider  now  surfaces  of  the  latter  kind. 

Let  S  be  such  a  surface  referred  to  its  lines  of  curvature,  and 
in  particular  let  the  lines  v  =  const,  be  spherical.  The  coordinates 
of  the  centers  of  the  spheres  as  well  as  their  radii  are  functions  of 
v  alone.  They  will  be  denoted  by  (V^  F2,  F3)  and  It.  By  Joachims- 
thal's  theorem  (§  59)  each  sphere  cuts  the  surface  under  the  same 
angle  at  all  its  points.  Hence  for  the  family  of  spheres  the  expres 
sion  for  the  angle  is  a  function  of  v  alone  ;  AVC  call  it  V. 

Since  the  direction-cosines  of  the  tangent  to  a  curve  u  =  const,  are 

dX  3Y  1     dZ' 


when  the  linear  element  of  the  spherical  representation  is  written 
do-2=  (odu2-}-  £dv\  the  coordinates  of  S  are  of  the  form 


/07\ 
(97) 


,   R  sin  VdX  , 
=  VA  ---  =-  —  +XR  cos  F, 


Tr_         T_ 

y  =  F2+  •—  +  YR  cos  F, 

7£  sin  F  dZ 


*  For  other  geomotrical  constructions  of  the  cyclides  of  Dupin  the  reader  is  referred 
to  the  article  in  the  Encyklopadie  der  Math.  Wissenschaflen,  Vol.  Ill,  3,  p.  290. 


SPHERICAL  LINES  OF  CURVATURE  315 

By  hypothesis  A,  lr,  Z  are  the  direction-cosines  of  the  normal  to  S ; 
consequently  we  must  have 

YA--  =  O,       VA--  =  O. 

^     du  ^     dv 

If  the  values  of  the  derivatives  obtained  from  (97)  be  reduced  by 
means  of  (V,  22),  and  the  results  substituted  in  the  above  equa 
tions,  the  first  vanishes  identically  and  the  second  reduces  to 

(98)  XV[  +  YV't  +  ZV'Z  +  (R  cos  V)' —  R  sin  FvV  =  0, 

where  the  primes  indicate  differentiation  with  respect  to  v.  Con 
versely,  when  this  condition  is  satisfied,  equations  (97)  define  a 
surface  on  which  the  curves  v  =  const,  are  spherical.  Hence : 

A  necessary  and  sufficient  condition  that  the  curves  v  =  const,  of  an 
orthogonal  system  on  the  unit  sphere  represent  spherical  lines  of  cur 
vature  upon  a  surface  is  that  five  functions  of  v,  namely  Vr  F2,  F3, 
R,  V,  can  I e  found  which  satisfy  the  corresponding  equation  (98). 

We  note  that  F1?  F2,  F3,  and  R  cos  V  are  determined  by  (98) 
only  to  within  additive  constants.  A  change  of  these  constants 
for  the  first  three  gives  a  translation  of  the  surface.  If  R  cos  V  be 
increased  by  a  constant,  we  have  a  new  surface  parallel  to  the 
other  one.  Hence  *  : 

If  the  lines  of  curvature  in  one  system  upon  a  surface  be  spherical, 
the  same  is  true  of  the  corresponding  system  on  each  parallel  surface. 

Since  equation  (98)  is  homogeneous  in  the  quantities  F/,  F^,  F3, 
(R  cos  F)',  R  sin  F,  the  latter  are  determined  only  to  within  a  factor 
which  may  be  a  function  of  v.  This  function  may  be  chosen  so 
that  all  the  spheres  pass  through  a  point.  From  these  results  we 
have  the  theorem  of  Dobriner  f  : 

With  each  surface  with  spherical  lines  of  curvature  in  one  system 
there  is  associated  an  infinity  of  nonparallel  surfaces  of  the  same 
kind  with  the  same  spherical  representation  of  these  lines  of  curvature. 
Among  these  surfaces  there  is  at  least  one  for  which  all  the  spheres 
pass  through  a  point.  At  corresponding  points  of  the  loci  of  the  cen 
ters  of  spheres  of  two  surfaces  of  the  family  the  tangents  are  parallel. 

*  Cf.  Bianchi,  Vol.  II,  p.  303.  f  Crelle,  Vol.  XCIV  (1883),  pp.  118,  125. 


316     SURFACES  WITH  PLANE  LINES  OF  CURVATURE 

If  the  values  of  x,  y,  z  from  (97)  be  substituted  in  the  formulas 
of  Rodrigues  (IV,  32), 

dx          dx       dx          dx 

<99)  £—"*'    a^-^' 

and  similarly  for  y  and  z,  we  obtain  by  means  of  (V,  22), 
—     =R  cos 


Conversely,  when  for  a  surface  referred  to  its  lines  of  curvature 
the  principal  radius  pl  is  of  the  form 

-».-*." 

where   (^   and  $2  are   any  functions  whatever  of  v,  the  curves 
v  =  const,  are  spherical.     For,  by  (V,  22), 


dv        V~       ov     du 


Consequently,  from  the  first  of  (99),  in  which  pl  is  given  the  above 
value,  we  obtain  by  integration 


where  Vl  is  a  function  of  v  alone.    Similar  results  follow  for  y  and  z. 
As  these  expressions  are  of  the  form  (97),  we  have  the  theorem  : 

A  necessary  and  sufficient  condition  that  the  lines  of  curvature 
v  =  const,  be  spherical  is  that  pl  be  of  the  form  (100). 


EXAMPLES 

1.  If  the  lines  of  curvature  in  one  system  are  plane  and  one  is  a  circle,  all 
are  circles. 

2.  When  the  lines  of  curvature  in  one  family  on  a  surface  are  circles,  their 
spherical  images  are  circles  whose  spherical  centers  constitute  the  spherical  indi- 
catrix  of  the  tangents  to  the  curve  of  centers  of  the  spheres  which  are  enveloped 
by  the  given  surface.    Show  also  that  each  one-parameter  system  of  circles  on  the 
unit  sphere  represents  the  circular  lines  of  curvature  on  an  infinity  of  surfaces, 
for  one  of  which  the  circles  are  equal. 


EXAMPLES  31T 

3.  If  the  lines  of  curvature  of  a  surface  are  parametric,  and  the  curves  u  =  const. 
are  spherical,  we  have  j  j  cot  F 

Pgu      B  sin  F        Pi 

where  pgu,  />i,  E  denote  the  radii  of  geodesic  curvature  and  normal  curvature  in  the 
direction  v  —  const,  and  of  the  sphere  respectively,  and  F  denotes  the  angle  under 
which  the  sphere  cuts  the  surface. 

4.  When  a  line  of  curvature  is  spherical,  the  developable  circumscribing  the 
surface  along  this  line  of  curvature  also  circumscribes  a  sphere ;  and  conversely, 
if  such  a  developable  circumscribes  a  sphere,  the  line  of  curvature  lies  on  a  sphere 
concentric  with  the  latter  (cf.  Ex.  7,  p.  149). 

5.  Let  S  be  a  pseudospherical  surface  with  the  spherical  representation  (25)  of 
its  lines  of  curvature.   Show  that  a  necessary  and  sufficient  condition  that  the  curves 
v  =  const,  be  plane  is  a  /    1     a&»\  _ 

du  \sin  w  dv/ 

show  also  that  in  this  case  w  is  given  by 

V'-U' 

COS  0)  =  — — , 


where  V  and  V  are  functions  of  u  and  v  respectively,  which  satisfy  the  conditions 

U'*  =  U"4  +  (a  -  2)  C72  +  6,         F'2  =  F*  +  aF2  -f  (a  -f  b  -  1), 
a  and  b  being  constants,  and  the  accent  indicating  differentiation,  unless  U'  or 
"V  is  zero. 

6.  When  the  lines  of  curvature  v  —  const,  upon  a  pseudospherical  surface  are 
plane,  the  linear  element  is  reducible  to  the  form 

_         a2  tanh2  (u  +  v)  dw2  a2  sech2  (u  4-  v}  dv2 

~  C  -A  cosh  2  u  -f  B  sinh  2  u      G  +  A  cosh  2  v  +  B  sinh  2  v  -  1 ' 
where  A,  B,  C  are  constants.    Find  the  expressions  for  the  principal  radii. 

7.  When  the  lines  of  curvature  v  =  const,  on  a  spherical  surface  are  plane,  the 
linear  element  is  reducible  to 

_  a2  cot2  (u  +  v)  dw2      a2  esc2  (u  +  v}  d i?2 
~  .A  sin  2  w  -f  B  -  I         A  sin  2  u  -  7? 

where  J.  and  J5  are  constants.    The  surfaces  of  Exs.  5  and  6  are  called  the  surfaces 
of  Enneper  of  constant  curvature. 


GENERAL  EXAMPLES 

1.  The  lines  of  curvature  and  the  asymptotic  lines  on  a  surface  of  constant 
curvature  can  be  found  by  quadratures. 

2.  When  the  linear  element  of  a  pseudospherical  surface  is  in  the  form  (iii)  of  (7), 

M 

the  equations  x  =  cw,  y  =  ae~a  determine  a  conformal  representation  of  the  surface 
upon  the  plane,  which  is  such  that  any  geodesic  on  the  surface  is  represented  on 
the  plane  by  a  circle  with  its  center  on  the  ic-axis,  or  by  a  line  perpendicular  to 
this  axis. 


318  TF-SURFACES 

3.  When  the  linear  elements  of  a  developable  surface,  a  spherical  surface,  and 
a  pseudospherical  surface  are  in  the  respective  forms 

ds~  =  du2  +  u-dv2,     ds'2  =  a?(du'2  +  sin2wdu2),     ds'2  =  «2(dw2  +  sinh^udw2), 
the  finite  equations  of  the  geodesies  are  respectively 

Au  cos  v  -f  Bu  sin  v  -f  C  =  0,     A  tan  u  cos  v  -f-  B  tan  u  sin  v  +  C  =  0, 

A  tanh  u  cos  v  +  .B  tanh  u  sin  v  +  C  =  0, 

where  A,  Z>,  C  are  constants ;  if  the  coefficients  of  A  and  B  are  in  any  case  equated 
to  x  and  y,  the  resulting  equations  define  a  correspondence  between  the  surface  and 
the  plane  such  that  geodesies  on  the  former  correspond  to  straight  lines  on  the  latter. 
Find  the  expression  for  each  linear  element  in  terms  of  x  and  y  as  parameters. 

4.  Each  surf  ace  of  center  of  a  pseudospherical  surf  ace  is  applicable  to  the  catenoid. 

5.  The  asymptotic  lines  on  the  surfaces  of  center  of  a  surface  of  constant  mean 
curvature  correspond  to  the  minimal  lines  on  the  latter. 

6.  Surfaces  of  constant  mean  curvature  are  characterized  by  the  property  that 
if  u  =  const. ,  v  =  const,  are  the  minimal  curves,  then  D  is  a  function  of  u  alone 
and  D"  of  v  alone. 

7.  Equation  (23)  admits  the  solution  w  =  0,  in  which  case  the  surface  degen 
erates  into  a  curve.    Show  that  the  general  integral  of  the  corresponding  equations 

M  +  r  cos  <r 

(35)  is  tan  0/2  =  Ce  Bin<r  ;  take  for  S  the  line  x  =  0,  y  -  0,  z  =  an  and  derive 
the  equations  of  the  transforms  of  -S;  shc^w  that  the  latter  are  surfaces  of  Dini 
(Ex.  C,  §  122),  or  a  pseudosphere. 

8.  Show  that  the  Backlund  transforms  of  the  surfaces  of  Dini  and  of  the  pseudo- 
sphere  can  be  found  without  integration,  and  that  if  the  pseudosphere  be  trans 
formed  by  the  transformation  of  Bianchi,  the  resulting  surface  may  be  defined  by 

2  a  cosh  u  2  a  cosh  u    . 

x  = —  (sinu  —  ucosu),         y— —  (cosv  +  vsm  v), 

V  V 


(2  sinh  u  cosh  u\ 
U        cosh2w  +  v2  /  ' 


Show  that  the  lines  of  curvature  v  —  const,  lie  in  planes  through  the  2-axis. 

9.  The  tangents  to  a  family  of  geodesies  of  the  elliptic  or  hyperbolic  type  on  a 
pseudospherical  surface  are  normal  to  a  W-surface  ;  the  relations  between  the  radii 

are  respectively  .  P\  +  c  .,  Pi  4-  c 

Pl  —  PO  =  a  tanh  —    — ,         pl  —  p2  =  a  coth  —   — , 
a  a 

where  a  and  c  are  constants  (cf.  §  7(3). 

10.  Show  that  the  linear  elements  of  the  second  surfaces  of  center  of  the 
>F-surfaces  of  Ex.  0  are  reducible  to  the  respective  forms 

ds.?  =  tanh4  -  du"2  +  sech2  -  dv2,         ds.?  -  coth4  U  du2  +  csch2  ~  du2, 
a  a  a  a 

and  that  consequently  these   surfaces  are  applicable  to  surfaces  of   revolution 
whose  meridians  are  defined  by 


a 

(log  tan 
Hog  tan 

0 

+ 
+ 

COS  0  j  , 
COS  0  j  , 

V* 

W  + 
a 

2 

Vi 

-  a2, 

2 

where  K  denotes  a  constant. 


GENERAL  EXAMPLES  319 

11.  Determine  the  particular  form  of  the  linear  element  (49),  and  the  nature 
of  the  curves  upon  the  surface  to  which  the  asymptotic  lines  on  the  sheets  of 
the  evolute  correspond,  when 

p.  11 

(a)      —  =  const;         (6)      --  —  =  const. 
Pz  Pi      Pz 

12.  When  a  JF-surface  is  of  the  type  (72),  the  surfaces  of  center  are  applicable 
to  one  another  and  to  an  imaginary  paraboloid  of  revolution. 

13.  When  a  IF-surface  is  of  the  type  (72)  and  the  linear  element  of  the  sphere 
has  the  form  (VI,  GO),  the  curves  u  +  v  —  const,  and  u  —  y  —  const,  on  the  spherical 
representation  are  geodesic  parallels  whose  orthogonal  trajectories  correspond  to 
the  asymptotic  lines  on  the  surfaces  of  center  ;  hence  on  each  sheet  there  is  a  family 
of  geodesies  such  that  the  tangents  at  their  points  of  meeting  with  an  asymptotic 
line  are  parallel  to  a  plane,  which  varies  in  general  with  the  asymptotic  line. 

14.  Show  that  the  equations 


-+  f  Vs'm-dv.        y  =  aUsin-  —  C 
a     J  a  a     J 


where  a  denotes  an  arbitrary  constant,  define  a  family  of  applicable  molding  surfaces. 

15.  When  the  lines  of  curvature  in  one  system  on  a  surface  are  plane,  and  the 
lines  of  the  second  system  lie  on  spheres  which  cut  the  surface  orthogonally,  the 
latter  is  a  surface  of  Joachimsthal. 

16.  The  spherical  lines  of  curvature  on  a  surface  of  Joachimsthal  have  constant 
geodesic  curvature,  the  radius  of  geodesic  curvature  being  the  radius  of  the  sphere 
on  which  a  curve  lies. 

17.  When  the  lines  of  curvature  in  one  system  on  a  surface  lie  on  concentric 
spheres,  it  is  a  surface  of  Monge,  whose  director-developable  is  a  cone  with  its 
vertex  at  the  center  of  the  spheres  ;  and  conversely. 

18.  The  sheets  of  the  evolute  of  a  surface  of  Monge  are  the  director-developable 
and  a  second  surface  of  Monge,  which  has  the  same  director-developable  and  whose 
generating  curve  is  the  evolute  of  the  generating  curve  of  the  given  surface. 

19.  If  the  lines  of  curvature  in  one  system  on  a  surface  are  plane,  and  two  in 
the  second  system  are  plane,  then  all  in  the  latter  system  are  plane. 

20.  A  surface  with  plane  lines  of  curvature  in  both  systems,  in  one  of  which 
they  are  circles,  is 

(a)  A  surface  of  Joachimsthal. 

(5)  The  locus  of  the  orthogonal  trajectories  of  a  family  of  spheres,  with  centers 
on  a  straight  line,  which  pass  through  a  circle  on  one  of  the  spheres. 

(c)  The  envelope  of  a  family  of  spheres  whose  centers  lie  on  a  plane  curve  C, 
and  whose  radii  are  proportional  to  the  distances  of  these  centers  from  a  straight 
line  fixed  in  the  plane  of  C. 

21.  If  an  arbitrary  curve  C  be  drawn  in  a  plane,  and  the  plane  be  made  to  move 
in  such  a  way  that  a  fixed  line  of  it  envelop  an  arbitrary  space  curve  T,  and  at  the 
same  time  the  plane  be  always  normal  to  the  principal  normal  to  T,  the  curve  C 
describes  a  surface  of  Monge. 


320  TF-SUKFACES 

22.  If  all  the  Bianchi  transforms  of  a  pseudospherical  surface  S  are  surfaces  of 
Enneper  (cf.  Ex.  5,  §  134),  S  is  a  surface  of  revolution. 

23.  When  u  has  the  value  in  Ex.  5,  §  134,  the  surfaces  with  the  spherical 
representation  (25),  and  with  the  linear  element 

ds*  =  (HI  cos  w  +  — Ydu2  +  U?  sin2  u  du2, 

where  U\  is  an  arbitrary  function  of  M,  are  surfaces  of  Joachimsthal. 

24.  If  the  lines  of  curvature  in  both  systems  be  plane  for  a  surface  S  with  the 
same  spherical  representation  of  its  lines  of  curvature  as  for  a  pseudospherical 
surface,  S  is  a  molding  surface. 

25.  If  S  is  a  pseudospherical  surface  with  the  spherical  representation  (25)  of 
its  lines  of  curvature,  and  the  curves  v  =  const,  are  plane,  the  function  6,  given  by 

„  c2w  „  aw  aw         .        aw       . 

sin  6  — -  +  cos  0 h  sin  w  —  =  0, 

a»2  aw  dv  dv 

determines  a  transformation  of  Bianchi  of  S  into  a  surface  Si  for  which  the  lines 
of  curvature  v  =  const,  are  plane. 

26.  A  necessary  and  sufficient  condition  that  the  lines  of  curvature  v=  const, 
on  a  pseudospherical  surface  with  the  representation  (25)  of  its  lines  of  curvature 
be  spherical  is  that  -\r     a., 

cotw  =  y1  +  J—  — , 

sin  w  dv 

where  V  and  V\  are  functions  of  v  alone.    Show  that  when  w  is  a  solution  of  (23) 
and  of 


i    aw  a  /   i    aw\ 

sin2  w  aw  aw  \sinwatv 

a  /   i    aw\  a2  /   i    aw\ 

au\sin2wau/  aM2\sinwau/ 


the  curves  v  =  const,  are  plane  or  spherical,  and  that  in  the  latter  case  V  and  V\ 
can  be  found  directly. 

27.  Show  that  when  w  is  a  solution  of  (23)  and  of 


dv  aucv2     a«2  awau     du  \au/ 

and  —  ( )  ^.  0,  the  lines  of  curvature  u  =  const,  are  spherical  on  the  pseudo- 

du  \cos  w  dv/ 

spherical  surface  with  the  spherical  representation  (25) ;  and  that  when  w  is  such 
a  function,  upon  the  surfaces  with  the  linear  element 


or 


/aw\2 

\dv/ 


/  r)w\2  I 

(sin  w  +  F—  )  du'2  +    cos  w  -f  V'+  V 
\  dv/ 


where  F  is  a  function  of  t>  alone,  the  curves  t>  =  const,  are  spherical;  in  the  former 
case  the  spheres  cut  the  surface  orthogonally. 


CHAPTER  IX 

DEFORMATION  OF  SURFACES 

135.  Problem  of  Minding.  Surfaces  of  constant  curvature.  Ac 
cording  to  §  43  two  surfaces  are  applicable  when  a  one-to-one 
correspondence  can  be  established  between  them  which  is  of 
such  a  nature  that  in  the  neighborhood  of  corresponding  points 
corresponding  figures  are  congruent  or  symmetric.  It  was  seen 
that  two  surfaces  with  the  same  linear  element  are  applicable, 
the  parametric  curves  on  the  two  surfaces  being  in  correspon 
dence.  But  the  fact  that  the  linear  elements  of  two  surfaces  are 
unlike  is  not  a  sufficient  condition  that  they  are  not  applicable ; 
in  evidence  of  this  we  have  merely  to  recall  the  effect  of  a  change 
of  parameters,  to  say  nothing  of  a  change  of  parametric  lines. 
Hence  we  are  brought  to  the  following  problem,  first  proposed 
by  Minding :  * 

To  find  a  necessary  and  sufficient  condition  that  two  surfaces  be 
applicable. 

From  the  second  theorem  of  §  64  it  follows  that  a  necessary 
condition  is  that  the  total  curvature  of  the  two  surfaces  at  corre 
sponding  points  be  the  same.  We  shall  show  that  this  condition 
is  sufficient  for  surfaces  of  constant  curvature. 

In  §  64  we  found  that  when  K  is  zero  at  all  points  of  a  surface, 
the  surface  is  applicable  to  the  plane.  If  the  plane  be  referred  to 
the  system  of  straight  lines  parallel  to  the  rectangular  axes,  its 

linear  element  is  797272 

ds2=dx2  +  dy*. 

Hence  the  analytical  problem  of  the  application  of  a  developable 
surface  upon  the  plane  reduces  to  the  determination  of  orthogonal 
systems  of  geodesies  such  that  when  these  curves  are  parametric 
the  linear  element  takes  the  above  form. 

*  Crelle,  Vol.  XIX  (1839),  pp.  371-387. 
321 


322  DEFORMATION  OF  SURFACES 

Referring  to  the  results  of  §  39,  we  see  that  in  this  case  the 
factor  ttv  must  equal  unity.  Consequently  we  must  find  a  function 
6  such  that  the  left-hand  members  of  the  equations 


du  +  -.dv   =  d(x  +  iy}, 
du  + 


\ 

are  exact  differentials,  in  which  case  these  equations  give  x  and  y 
by  quadratures.    Hence  we  must  have 


du\        -^E  99 

which  are  equivalent  to 


du~Hu      ZEHdu      2  H  dv 

dd  _  J_  d_G_  _      F     d_E_ 
'dv  ~  ^H  ~du      2  EH  dv  ' 

From  (V,  12)  it  is  seen  that  these  equations  are  consistent  when 
K=  0.  In  this  case  6,  and  consequently  x  and  y,  can  be  found  by 
quadratures. 

The  additive  constants  of  integration  are  of  such  a  character 
that  if  ar0,  y0  are  a  particular  set  of  solutions,  the  most  general  are 

x  =  x0  cos  a  —  yQ  sin  a  +  a,         y  =  XQ  sin  a  +  yQ  cos  a  +  />, 

where  a,  #,  5  are  arbitrary  constants. 

In  the  above  manner  we  can  effect  the  isometric  representation 
of  any  developable  surface  upon  the  plane,  and  consequently  upon 
itself  or  any  other  developable.  These  results  may  be  stated  thus  : 

A  developable  surface  is  applicable  to  itself,  or  to  any  other  develop 
able,  in  a  triple  infinity  of  ways,  and  the  complete  determination  of 
the  applicability  requires  quadratures  only. 

Incidentally  we  have  the  two  theorems: 

The  geodesies  upon  a  developable  surface  can  be  foundby  quadratures. 
If  the  total  curvature  of  a  quadratic  form  be  zero,  the  quadratic 
form  is  reducible  by  quadratures  to  dad&. 


SUEFACES  OF  CONSTANT  CURVATURE      323 

Suppose  now  that  the  total  curvature  of  two  surfaces  S,  Sl  is 
I/a2,  where  a  is  a  real  constant.  Let  P  and  7?  be  points  on  S 
and  /S^  respectively,  C  and  Cl  geodesies  through  these  respective 
points,  and  take  P  and  I[  for  the  poles  and  C  and  Cl  for  the 
curves  v  =  0  of  a  polar  geodesic  system  on  these  surfaces.  The 
linear  elements  are  accordingly  (VIII,  6) 

d**  =  du2  +  sin2  -  dv\         ds2  =  du*  +  sin2  ^  dv2. 
Hence  the  equations       u  —u  v  =  ±  v 

determine  an  isometric  representation  of  one  surface  upon  the 
other,  in  which  P  and  C  correspond  to  P  and  Cl  respectively. 
According  as  the  upper  or  lower  sign  in  the  second  equation 
is  used,  corresponding  figures  are  equal  or  symmetric.  Similar 
results  obtain  for  pseudospherical  surfaces.  Hence  we  have: 

Any  two  surfaces  of  constant  curvature,  different  from  zero,  are  in 
two  ways  applicable  so  that  a  given  point  and  geodesic  through  it  on  one 
surface  correspond  to  a  given  point  and  geodesic  through  it  on  the  other. 

In  particular,  a  surface  of  constant  curvature  can  be  applied  to 
itself  so  that  a  given  point  shall  go  into  any  other  point  and  a 
geodesic  through  the  former  into  one  through  the  latter.  Combin 
ing  these  results  with  the  last  theorem  of  §  117,  we  have: 

A  nondevelopable  surface  of  constant  curvature  can  be  applied  to 
itself,  or  to  any  surface  of  the  same  curvature,  in  a  triple  infinity  of 
ways,  and  the  complete  realization  of  the  applicability  requires  the 
solution  of  a  Iliccati  equation. 

136.  Solution  of  the  problem  of  Minding.    We  proceed  to  the 
determination  of  a  necessary  and  sufficient  condition  that  two  sur 
faces  S,  8'  of  variable  curvature  be  applicable.    Let  their  linear 
elements  be 
ds2  =  E  du2  +  2  Fdudv  +  G  dv2,    ds'2  =  E'  du''2  +  2F'  du'dv'  +  G'  dv'2. 

By  definition  S  and  S'  are  applicable  if  there  exist  two  independ 
ent  equations 

(1)  (/>  (U,   V)  =  $(UJ,  V1),  ^  (U,  V)  =  ^<(U',  V'), 

establishing  a  one-to-one  correspondence  between  the  surfaces  of 
such  a  nature  that  by  means  of  (1)  either  of  the  above  quadratic 
forms  can  be  transformed  into  the  other. 


324  DEF011MATION  OF  SURFACES 

It  is  evident  that  if  the  two  surfaces  are  applicable,  the  differen 
tial  parameters  formed  with  respect  to  the  two  linear  elements  are 
equal.  Hence  a  necessary  condition  is 

(2)       A^  =  A;</>',   A^,  f)=A!(<#>',  f),   A1f=A;t', 

where  the  primes  indicate  functions  pertaining  to  S'.  These  con 
ditions  are  likewise  sufficient  that  the  transformation  (1)  change 
either  of  the  above  quadratic  forms  into  the  other.  For,  if  the 
curves  (/>  =  const.,  ^r  —  const.  ;  </>'  =  const.,  ^'  =  const,  be  taken  for 
the  parametric  curves  on  S  and  Sf  respectively,  the  respective 
linear  elements  may  be  written  (cf.  §  37) 
, 

„ 

Hence  when  equations  (1)  and  (2)  hold,  the  surfaces  are  applicable. 
The  next  step  is  the  determination  of  equations  of  the  form  (1). 
Since  the  curvature  of  two  applicable  surfaces  at  corresponding  points 
is  the  same,  one  such  equation  is  afforded  by  the  necessary  condition 
(3)  K(u,v)  =  K'(u',v'). 

The  first  of  equations  (2)  is 

(4)  A^A;*'. 

Both  members  of  this  equation  cannot  vanish  identically.  For,  in 
this  case  the  curves  K  —  const,  and  K'  =  const,  would  be  minimal 
(§  37),  and  consequently  imaginary.  If  these  two  equations  are 
independent  of  one  another,  that  is, 


they  establish  a  correspondence,  and  the  condition  that  it  be  iso 
metric  is,  as  seen  from  (2), 


If,  however, 
(5)  \K 

we  may  take  for  the  second  of  (1) 
(6) 

unless 
(7)  A2JfiT 


PROBLEM  OF  MINDING  325 

If  this  condition  be  not  satisfied,  the  conditions  that  (3),  (6)  define 
an  isometric  correspondence  are 


A1AaJST  = 

Finally,  we  consider  the  case  where  both  (5)  and  (7)  hold.  Since 
the  ratio  of  \K  and  A2  K  is  a  function  of  JC,  the  curves  K  =  const. 
and  their  orthogonal  trajectories  t  =  const,  form  an  isothermal  sys 
tem  of  lines  on  S  (§  41).  Moreover,  the  function  t  can  be  found 
by  quadratures,  and  the  linear  element  is  reducible  to 


- 

(8)  ds2  =  -—  (dK2  +  eJ  '<A'>     dt2). 

J(K) 

When  in  particular  A2^T=  0,  the  linear  element  is 


In  like  manner  the  linear  element  of  S'  is  reducible  to 


or,  in  the  particular  case  A^'  =  0,  to 


In  either  case  the  equations 

K  =  K'i       t  =  ±t'+a, 

where  a  is  an  arbitrary  constant,  define  the  applicability  of  the 
surfaces. 

We  have  thus  treated  all  possible  cases  and  found  that  it  can 
be  determined  without  quadrature  whether  two  surfaces  are  appli 
cable.  Moreover,  in  the  first  two  cases  the  equations  defining  the 
correspondence  follow  directly,  but  in  the  last  case  the  determina 
tion  requires  a  quadrature.  The  last  case  differs  also  in  this  respect  : 
the  application  can  be  effected  in  an  infinity  of  ways,  whereas  in 
the  first  two  cases  it  is  unique. 

*  If  the  surface  be  referred  to  the  curves  <r  =  const,  and  their  orthogonal  trajectories, 

where  a  —  C—=  ,  equation  (6)  may  be  replaced  by  A2tr  =  A^',  and  it  can  be  shown 

J  ~vf(K) 

that  AI(<T,  &<£<?)  =  Ai(<r',  A%<r')  is  a  consequence  of  the  other  conditions.     Cf.  Darboux, 
Vol.  Ill,  p.  227. 


326  DEFORMATION  OF  SURFACES 

Furthermore,  we  notice  from  (8)  that  in  the  third  case  the  sur 
face  S  is  applicable  to  a  surface  of  revolution,  the  parallels  of  the 
latter  corresponding  to  the  curves  K=  const,  of  the  former.  Con 
versely,  the  linear  element  of  every  surface  applicable  to  a  surface 
of  revolution  can  be  put  in  the  form  (8).  For,  a  necessary  and 
sufficient  condition  that  a  surface  be  applicable  to  a  surface  of 
revolution  is  that  its  linear  element  be  reducible  to 


where  U  is  a  function  of  u  alone  (§  46).    Now 

U"  Cdu 

AlW  =  l,         X=    --, 

From   the   second   it   follows  that   u  —  F(K),   and   consequently 
—  —F^K).    When  these  values  are  substituted  in  the  above 
equations,  we  have,  in  consequence  of  Ex.  5,  p.  91, 
(9)  A1/C=/(A"),         A,JiT  =*<*). 

Hence  we  have  the  theorem  : 

Equations  (9)  constitute  a  necessary  and  sufficient  condition  that  a 
surface  be  applicable  to  a  surface  of  revolution. 

The  equations 

K=K,          t  =  ±t'+a 

define  an  isometric  representation  of  a  surface  with  the  linear  ele 
ment  (8)  upon  itself.    Therefore  we  have  : 

Every  surface  applicable  to  a  surface  of  revolution  admits  of  a 
continuous  deformation  into  itself  in  such  a  way  that  each  curve 
K  =  const,  slides  over  itself. 

Conversely,  every  surface  applicable  to  itself  in  an  infinity  of 
ways  is  applicable  to  a  surface  of  revolution.  For,  if  the  curvature 
is  constant,  the  surface  is  applicable  to  a  surface  of  revolution 
(§  135),  and  the  only  case  in  which  two  surfaces  of  variable  curva 
ture  are  applicable  in  an  infinity  of  ways  is  that  for  which  condi 
tions  (5)  and  (7)  are  satisfied. 


DEFORMATION  OF  MINIMAL  SURFACES  327 

137.  Deformation  of  minimal  surfaces.  These  results  suggest  a 
means  of  determining  the  minimal  surfaces*  applicable  to  a  surface 
of  revolution.  In  the  first  place  we  inquire  under  what  conditions 
two  minimal  surfaces  are  applicable.  The  latter  problem  reduces 
to  the  determination  of  two  pairs  of  parameters,  w,  v  and  u^  vv  and 
two  pairs  of  functions,  F(u),  <f>(v)*  and  Ffa^,  ^>1(f1),  which  satisfy 
the  condition 

(10)  (1  +  uvfF(u)®(v)  dudv  =  (1  +  u^)*  f\(uj  4^)  du^dvr 

From  the  nature  of  this  equation  it  follows  that  the  equations  which 
serve  to  establish  the  correspondence  between  the  two  surfaces  are 
either  of  the  form 

(11)  ^=0(M),  V1=^(V), 

or 

(12)  «,=  *(»),          »,  =  *(«)• 

If  either  set  of  values  for  ul  and  vl  be  substituted  in  (10),  and  if 
after  removing  the  common  factor  dudv  we  take  the  logarithmic 
derivative  with  respect  to  u  and  v,  we  obtain 


(1  +  u^Y      (1  +  uvf 
As  this  may  be  written 
,..  o  duldvl  dudv 

(i  +  uft)*  ~~  (I  +^^y2  ' 

the  spherical  images  of  corresponding  parts  on  the  two  surfaces  are 
equal  or  symmetric  according  as  (11)  or  (12)  obtains  (§  47).  The 
latter  case  reduces  to  the  former  when  the  sense  of  the  normal  to 
either  surface  is  changed.  When  this  has  been  done,  corresponding 
spherical  images  are  equal  and  can  be  made  to  coincide  by  a  rota 
tion  of  the  unit  sphere  about  a  diameter.  Hence  one  surface  can  be 
so  displaced  in  space  that  corresponding  normals  become  parallel, 
in  which  case  the  two  surfaces  have  the  same  representation,  that 
is,  Wj  =  u,  vx=  v.  Now  equation  (10)  is 


which  is  equivalent  to 

Fl(u)=cF(u),          3 

*  §  no. 


328  DEFORMATION  OF  SURFACES 

where  c  denotes  a  constant.  If  the  surfaces  are  real,  c  must  be  of 
the  form  ei<x.  Hence,  in  consequence  of  §  113,  we  have  the  theorem : 
A  minimal  surface  admits  of  a  continuous  deformation  into  an 
infinity  of  minimal  surfaces,  which  are  either  associate  to  it  or  can 
be  made  such  by  a  suitable  displacement. 

We  pass  to  the  determination  of  a  minimal  surface  which  admits 
of  a  continuous  deformation  into  itself,  and  consequently  is  appli 
cable  to  a  surface  of  revolution.  In  consequence  of  the  interpre 
tation  of  equation  (13)  it  follows  that  if  a  minimal  surface  be 
deformed  continuously  into  itself,  a  point  p  on  the  sphere  tends  to 
move  in  the  direction  of  the  small  circle  through  p,  whose  axis  is 
the  momentary  axis  of  rotation,  and  consequently  each  of  these 
small  circles  moves  over  itself.  From  §  47  it  follows  that  if  the 
axis  of  rotation  be  taken  for  the  2-axis,  these  small  circles  are  the 
curves  uv  =  const.  In  the  deformation  each  point  of  the  surface 
moves  along  the  curve  K=  const,  through  it.  Hence  K  is  a  func 
tion  of  uv.  From  (VII,  100,  102)  we  have 

A=___^l_; 

consequently  F(u)Q(v)  must  be  a  function  of  uv,  and  hence 

uF'(u)  __  v<&'(v) 
•       F(u)         4>(v) 

The  common  value  of  these  two  terms  is  a  constant.     If  it  be 
denoted  by  K,  we  have  K, 

where  c  and  c1  are  constants.    Hence  from  (VII,  98)  we  have : 

Any  minimal  surface  applicable  to  a  surface  of  revolution  can  be 
defined  by  equations  of  the  form 

*  c  f  (1  _  u*)  u«du  +  |  <?j  f  (1  -  v1)  v*dv, 

-c  Cfl  +  u^u'du  —  ^Ci  1(1 
2   J  {  2V 

/r  <+i 
u'^du  +  c^Ji       iv, 

ivhere  c,  c.,  and  K  are  arbitrary  constants. 


(14) 


y 


DEFOKMATION  OF  MINIMAL   SUEFACES  329 

Since  the  curves  K  —  const,  are  represented  on  the  sphere  by  the 
small  circles  whose  axis  is  the  z-axis,  in  each  finite  deformation  of 
the  surface  into  itself,  as  well  as  in  a  very  small  one,  the  unit  sphere 
undergoes  a  rotation  about  this  axis.  In  §  47  it  was  seen  that  such 
a  rotation  is  equivalent  to  replacing  u,  v  by  ueia,  ve~ia,  where  a 
denotes  the  angle  of  rotation.  Hence  the  continuous  deformation 
of  a  surface  (14)  is  defined  by  the  equations  resulting  from  the 
substitution  in  (14)  of  ueia,  ve~i<z  for  u,  v  respectively. 

An  important  property  of  the  surfaces  (14)  is  discovered  when 
we  submit  such  a  surface  to  a  rotation  of  angle  a  about  the 
z-axis.  Let  S  denote  the  surface  in  its  new  position,  and  write 
its  equations  in  the  form 


f  (1  -  tf 


and  similarly  for  y  and  z.  Between  the  parameters  u,  v  and  u,  v 
the  following  relations  hold: 

u  =  ueta,         v  =  ve~  tar, 
and  we  have  also 

x  =  x  cos  a  —  y  sin  a,          ~y  =  x  sin  a  -f  y  cos  a:,          z  =  z. 
Combining  these  equations  with  (14),  we  find 

F  (u)  =  cuKe~  ia(K  +  2),          5>  (v)  =  c{vKeia(K  +  2). 

Hence,  for  the  correspondence  defined  by  u  =  ?/,  v  =  v,  the  surface  S 
is  an  associate  of  S,  unless  K  -f-  2  =  0,  in  which  case  it  is  the  same 
surface. 

We   consider   the   latter   case,   and   remark   that   its    equations 
are  (cf.  §  110) 


If  u,  v  be  replaced  by  ueia,  ve~ia,  and  the  resulting  expressions  be 
denoted  by  xv  y^  zx,  we  have 


(15)    xl  =  xcosa  —  ysina1    y^  =  x  sin  a  +  y  cos  a,    zl  =  z  +  clR(iac}. 

Hence,  in  a  continuous  deformation,  the  surface  slides  over  itself 
with  a  helicoidal  motion.  Consequently  it  is  a  helicoid.  Moreover, 
it  is  the  only  minimal  helicoid.  For,  every  helicoid  is  applicable 


330  DEFORMATION  OF  SURFACES 

to  a  surface  of  revolution,  and  each  minimal  surface  applicable  to 
a  surface  of  revolution  with  the  z-axis  for  the  axis  of  revolution 
of  the  sphere  is  defined  by  (14).  But  only  when  ic  =  —  2  will  the 
substitution  of  we1'",  ve~ia  give  a  set  of  equations  such  as  (15). 
Hence  we  have  : 

The  helicoidal  minimal  surfaces  are  defined  by  the  Weierstrass 
formulas  when  F(u)=c/u2. 

And  we  may  state  the  other  results  thus  : 

If  any  nonhelicoidal  minimal  surface,  which  is  applicable  to  a 
surface  of  revolution,  be  rotated  through  any  angle  about  the  axis  of 
the  unit  sphere  whose  small  circles  represent  the  curves  K  —  const. 
on  the  surface,  and  a  correspondence  with  parallelism  of  tangent 
planes  be  established  between  the  surfaces,  they  are  associate  ;  con 
sequently  the  associates  of  such  a  minimal  surface  are  supcrposable. 

EXAMPLES 

1  .  Find  under  what  conditions  the  surfaces,  whose  equations  are 


z\  —  F(r)  +  av, 

can  be  brought  into  a  one-to-one  correspondence,  so  that  the  total  curvature  at 
corresponding  points  is  the  same.  Determine  under  what  condition  the  surfaces 
are  applicable. 

2.  If  the  tangent  planes  to  two  applicable  surfaces  at  corresponding  points  are 
parallel,  the  surfaces  are  associate  minimal  surfaces. 

3.  Show  that  the  equations 

x  =  eaut        y  =  e-  av,        z  —  aeauz  -f  b<y~  av2, 

where  a  is  a  real  parameter,  and  a  and  6  are  constants,  define  a  family  of  parab 
oloids  which  have  the  same  total  curvature  at  points  with  the  same  curvilinear 
coordinates.  Are  these  surfaces  applicable  to  one  another  ? 

4.  Find  the  geodesies  on  a  surface  with  the  linear  element 

duz  _  4  y  dudv  +  4  u  dv2 
ds2  =.  -- 
4(w-t>2) 

Show  that  the  surface  is  applicable  to  a  surface  of  revolution,  and  determine  the 
form  of  a  meridian  of  the  latter. 

5  .  Determine  the  values  of  the  constants  a  and  6  in 

ds*  =  du2  +  [(u  +  au)2  +  62]  du2, 

so  that  a  surface  with  this  linear  element  shall  be  applicable  to 
(a)  the  right  helicoid. 
(&)  the  ellipsoid  of  revolution. 


SECOND  GENERAL  PROBLEM          331 

6.  A  necessary  and  sufficient  condition  that  a  surface  be  applicable  to  a  surface 
of  revolution  is  that  each  curve  of  a  family  of  geodesic  parallels  have  constant 
geodesic  curvature. 

7.  Show  that  the  helicoidal  minimal  surfaces  are  applicable  to  the  catenoid  and 
to  the  right  helicoid. 

138.  Second  general  problem  of  deformation.  We  have  seen  that 
it  can  always  be  determined  whether  or  not  two  given  surfaces  are 
applicable  to  one  another.  The  solution  of  this  problem  was  an 
important  contribution  to  the  theory  of  deformation.  An  equally 
important  problem,  but  a  more  difficult  one,  is  the  following : 

To  determine  all  the  surfaces  applicable  to  a  given  one. 

This  problem  was  proposed  by  the  French  Academy  in  1859,  and 
has  been  studied  by  the  most  distinguished  geometers  ever  since. 
Although  it  has  not  been  solved  in  the  general  case,  its  profound  study 
has  led  to  many  interesting  results,  some  of  which  we  shall  derive. 

If  the  linear  element  of  the  given  surface  be 

c?s2  =  Edu?  +  2  F  dudv  -f-  G  dv*, 

every  surface  applicable  to  it  is  determined  by  this  form  and  by  a 
second,  namely  Ddu*+  2  D'dudv  +  D"dv2,  whose  coefficients  satisfy 
the  Gauss  and  Codazzi  equations  (§  64).  Conversely,  every  set  of 
solutions  D,  D',  D"  of  these  equations  defines  a  surface  applicable 
to  the  given  one,  and  the  determination  of  the  Cartesian  coordinates 
of  the  corresponding  surface  requires  the  solution  of  a  Riccati  equa 
tion.  But  neither  the  Codazzi  equations,  nor  a  Riccati  equation,  can 
be  integrated  in  the  general  case  with  our  present  knowledge  of 
differential  equations.  Later  we  shall  make  use  of  this  method  in 
the  study  of  particular  cases,  but  for  the  present  we  proceed  to 
the  exposition  of  another  means  of  attacking  the  general  problem. 
When  the  values  of  D,  D',  Dn  obtained  from  the  Gauss  equations 
(V,  7)  are  substituted  in  the  equation  H2K=J)D"—D'2,  the  result 
ing  equation  is  reducible,  inconsequence  of  the  identity  A^  =1— A'2 
(cf.  Ex.  6,  p.  120),  to 

(16) 


dtf 

"  \du dv~\  1  J  ^      I  2  J'dv 


332  DEFORMATION  OF  SURFACES 

This  equation,  which  is  satisfied  also  by  y  and  2,  involves  only 
E,  F,  G  and  their  derivatives,  and  consequently  its  integration 
will  give  the  complete  solution  of  the  problem.  It  is  linear  in 

\tfxtfx      /  c'2x  Yl     &x      tfx      tfx         ,  ,  ,       ,  ,  ^,      - 

^ri^—l^r-sr)    '  oTi1  T-T-'  Tl'  and  therefore  is  of  the  form 
\_du  cv       \dudv  /  J     dir     ^<7v     0tr 

studied  by  Ampere.    Hence  we  have  the  theorem: 

The  determination  of  all  surfaces  applicable  to  a  given  one  requires 
the  integration  of  a  partial  differential  equation  of  the  second  order 
of  the  Ampere  type. 

In  consequence  of  (16)  and  (V,  36)  we  have  that  the  coordinates 
of  a  surface  with  the  linear  element 

(17)  ds2  =  Edu2  +  2  Fdudv  +  G  dv2 
are  integrals  of 

(18)  A220  =  (l—  A^JST, 

the  differential  parameters  being1  formed  with  respect  to  (17).  We 
shall  find  that  when  one  of  these  coordinates  is  known  the  other 
two  can  be  found  by  quadratures. 

Our  general  problem  may  be  stated  thus  : 

G-iven  three  functions  E,  F,  G  of  u  and  v  ;  to  find  all  functions 
x,  y,  z  of  u  and  v  which  satisfy  the  equation 

dx2  +  dy1  +  dz2  =  E  du*  +  2  Fdudv  +  G  dv\ 
where  du  and  dv  may  be  chosen  arbitrarily. 

Darboux  *  observed  that  as  the  equation  may  'be  written 

(19)  dx'2+  dy*  =  Edu2+2Fdudv  +  Gdv2-  dz\ 

whose  left-hand  member  is  the  linear  element  of  the  plane,  or  of  a 
developable  surface,  the  total  curvature  of  the  quadratic  form 


(20)       \E-  p?Y"U'+  2\F-  *  **\ 
L         WJ  L         dudv\ 


dudv 
dv\ 

must  be  zero  (§  64). 

In  order  to  find  the  condition  for  this,  we  assume  that  z  is 
known,  and  take  for  parametric  lines  the  curves  z  =  const,  and  their 

*L.c.,  Vol.  Ill,  p.  253. 


SECOND  GENERAL  PROBLEM          333 

orthogonal  trajectories  for  v  =  const.  With  this  choice  of  parame 
ters  the  right-hand  member  of  (19)  reduces  to  (E—  I)dz2  +  Gdv*. 
The  condition  that  the  curvature  of  this  form  be  zero  is 


tr 


where  K  denotes  the  curvature  of  the  surface.  But  this  is  the 
condition  also  that  z  be  a  solution  of  (18)  when  the  differential 
parameters  are  formed  with  respect  to  Edz'2-}-Gdv2.  However,  the 
members  of  equation  (18)  are  differential  parameters  ;  consequently 
z  is  a  solution  of  this  equation  whatever  be  the  parametric  curves. 
By  reversing  the  above  steps  we  prove  the  theorem  : 

When  z  is  any  integral  of  the  equation  (18),  the  quadratic  form  (20) 
has  zero  curvature. 

When  such  a  solution  is  known  we  can  find  by  quadratures 
(cf.  §  135)  two  functions  x,  y  such  that  the  quadratic  form  (20)  is 
equal  to  dx*  +  dyz,  provided  that 


that  is,  Ajg  <  1.    Hence  we  have  the  theorem  : 

If  z  be  a  solution  of  A220  —  (1  —  AX0)  K  such  that  A^  <  1,  it  is  one 
of  the  rectangular  coordinates  of  a  surface  with  the  given  linear  ele 
ment,  and  the  other  two  coordinates  can  be  obtained  by  quadratures. 

139.  Deformations  which  change  a  curve  on  the  surface  into  a 
given  curve  in  space.  We  consider  the  problem  : 

Can  a  surface  be  deformed  in  such  a  manner  that  a  given  curve  C 
upon  it  comes  into  coincidence  with  a  given  curve  F  in 


Let  the  surface  be  referred  to  a  family  of  curves  orthogonal  to  C 
and  to  their  orthogonal  trajectories,  C  being  the  curve  v  =  0,  and 
its  arc  being  the  parameter  u,  so  that  E  —  \  for  v  =  0.  The  same 
conditions  hold  for  F  on  the  deform. 


334  DEFORMATION  OF  SURFACES 

Since  the  geodesic  curvature  of  C  is  unaltered  in  the  deformation 
(§  58),  it  follows  from  the  equation  (IV,  47)  for  the  new  surface, 
namely 
(21)  p  =  pgamw, 

that  the  deformation  is  impossible,  if  the  curvature  of  F  at  any 
point  is  less  than  the  geodesic  curvature  of  C  at  the  corresponding 
point.  Since  both  p  and  pg  are  known,  equation  (21)  determines 
to,  and  consequently  the  direction  of  the  normal  to  the  new  surface 
along  F  is  fixed.  This  being  the  case,  the  direction  of  the  tangents 
to  the  curves  u  =  const,  on  the  new  surface  at  points  of  F  can  be 

.    1    dx     1    c>f      1    cz  £ 

found,  and  so  we  have  the  values  of  — —  — »  -— —  r-i  —=  —  tor  v  =  U, 

V<?  to   \IG  co   vV;  w 

as  well  as  — »  —  >  —  for  v  =  0,  the  latter  being  the  direction-cosines 
cu    cu    du 

of  the  tangent  to  F.    If  these  expressions  be  differentiated  with 

,  tfx    c~y    d2z      c2x      tfy       c2z 

respect  to  u,  we  obtain  the  values  ot  — »  — ^»  7—7, ;  — — »  T— r-  >  7—7- 

3u"    cu"    cu"     cudv    ducv    cucv 

for  v=0.    Since  F=Q  and  E  =  1  for  v  =  0,  the  Gauss  equations 

(V,  7)  for  v  =  0  are 

tfx  1    SEdx 

—  =  — r  JJJL+ 

du2          2G  to  to 


J.     I/ Jjy     t/*f  J-  *y*-*      f**-' 

"  2  "a^  a^      2G  du  to 


dv*          2  cu  du      2  6r  00  dv 

All  the  terms  of  the  first  two  equations  have  been  determined 
except  D  and  D1 ;  hence  the  latter  are  given  by  these  equations. 
Since  the  total  curvature  A"  is  unaltered  by  the  deformation,  it  is 
known  at  all  points  of  F;  consequently  //'is  given  by  H*K  = 
/>/>"  — />'2,  unless  D  is  zero,  in  which  case  F  is  an  asymptotic  line 

d'2x 
and  p  =p(J.    When  /)"  is  found  we  can  obtain  the  value  of  — 5  from 

the  last  of  equations  (22).  From  the  method  of  derivation  of  equa 
tion  (16)  it  follows  that  the  above  process  is  equivalent  to  finding 

the  value  of  ^  from  equation  (16),  which  is  possible  unless  D  =  0. 

to 
Excluding  this  exceptional  case,  we  remark  that  if  equations  (22) 


PARTICULAR  DEFORMATIONS  335 

be  differentiated  with  respect  to  u,  we  obtain  the  values  of  all  the 
derivatives  of  x  of  the  third  order  for  v  =  0  except  —  -•  The  latter 

may  be  obtained  from  the  equation  which  results  from  the  differ 
entiation  of  equation  (16)  with  respect  to  v.  By  continuing  this 
process  we  obtain  the  values  for  v  —  0  of  the  derivatives  of  x  of  all 
orders,  and  likewise  of  y  and  z.  If  we  indicate  by  subscript  null 
the  values  of  functions,  when  u  =  UQ,  v  =  0,  the  expansions 

fdx\         idx 
=  z  +   — 


n 
\dufy       \dvfy       2\#ir/o        \dudvh 

and  similar  expansions  for  y  and  2,  are  convergent  in  general,  as 
Cauchy  has  shown,*  and  x,  y,  z  thus  defined  are  the  solutions  of 
equation  (16)  which  for  v  =  0  satisfy  the  given  conditions.  Hence  : 

A  surface  S  can  be  deformed  in  such  a  manner  that  a  curve  C  upon 
it  comes  into  coincidence  with  a  given  curve  F,  provided  that  the 
curvature  of  F  at  each  point  is  greater  than  the  geodesic  curvature 
of  C  at  the  corresponding  point. 

There  remains  the  exceptional  case  p  =  pa.  If  the  desired  def 
ormation  is  possible,  F  is  an  asymptotic  line  on  the  deform,  and 
consequently,  by  Enneper's  theorem  (§  59),  its  radius  of  torsion 
must  satisfy  the  condition  r2  =  —  1/JC.  Hence  when  C  is  given,  F 
is  determined,  if  it  is  to  be  an  asymptotic  line. 

If  F  satisfies  these  conditions,  the  value  of  D"  for  v  =  0  is  arbi 
trary,  as  we  have  seen.  But  when  it  has  been  chosen,  the  further 
determination  of  the  values  of  the  derivatives  of  #,  y,  z  of  higher 
order  for  v  =  0  is  unique,  it  being  the  same  as  that  pursued  in 
the  general  case.  Hence  equation  (16)  admits  as  solution  a  family 
of  these  surfaces,  depending  upon  an  arbitrary  function.  For  all 
.of  these  surfaces  the  directions  of  the  tangent  planes  at  each 
point  of  F  are  the  same.  Hence  we  have  the  theorem  : 

Criven  a  curve  C  upon  a  surface  8  ;  there  exists  in  space  a  unique 
curve  F  with  which  C  can  be  brought  into  coincidence  by  a  deforma 
tion  of  8  in  an  infinity  of  ways  ;  moreover,  all  the  new  surfaces  are 
tangent  to  one  another  along  F. 

*  Cf  .  Goursat,  Lemons  surT  integration  des  Equations  aux  derivees  partielles  du  second 
ordre,  chap.  ii.  Paris,  1896. 


336  DEFOKMATION  OF  SURFACES 

If  C  is  an  asymptotic  line  on  S,  it  may  be  taken  for  F;  hence : 
A  surface  may  be  subjected  to  a  continuous  deformation  during 

which  a  given  asymptotic  line  is  unaltered  in  form  and  continues  to 

be  an  asymptotic  line  on  each  deform. 

This  result  suggests  the  problem : 

Can  a  surface  be  subjected  to  a  continuous  deformation  in  which  a 
curve  other  than  an  asymptotic  line  is  unaltered? 

By  hypothesis  the  curvature  is  not  changed  and  the  geodesic 
curvature  is  necessarily  invariant;  hence  from  (21)  we  have  that 
sin  o>  must  have  the  same  value  for  all  the  surfaces.  If  o>  is  the 
same  for  all  surfaces,  the  tangent  plane  is  the  same,  and  consequently 
the  expansions  (23)  are  the  same.  Hence  all  the  surfaces  coincide 
in  this  case.  However,  there  are  always  two  values  of  W  for  which 
sin  o>  has  the  same  value,  unless  w  is  a  right  angle.  Hence  it  is 
possible  to  have  two  applicable  surfaces  passing  through  a  curve 
whose  points  are  self-correspondent,  but  not  an  infinity  of  such 
surfaces.  Therefore : 

An  asymptotic  line  is  the  only  curve  on  a  surface  which  can  remain 
unaltered  in  a  continuous  deformation. 

140.  Lines  of  curvature  in  correspondence.  We  inquire  whether 
a  surface  S  can  be  deformed  in  such  a  manner  that  a  given  curve 
C  upon  it  may  become  a  line  of  curvature  on  the  new  surface. 
Suppose  it  is  possible,  and  let  F  denote  this  line  of  curvature. 
The  radii  of  curvature  and  torsion  of  F  must  satisfy  (21)  and 
1/T  —  dto/ds  —  0  (cf.  §  59),  where  pg  is  the  same  ;for  F  as  for  C.  If 
we  choose  for  w  any  function  whatever,  the  functions  p  and  r  are 
thus  determined,  and  F  is  unique.  Since  o>  fixes  the  direction  of  the 
tangent  plane  to  the  new  surface  along  F,  there  is  only  one  deform 
of  S  of  the  kind  desired  for  each  choice  of  w  (cf.  §  139).  Hence : 

A  surface  can  be  deformed  in  an  infinity  of  ways  so  that  a  given 
curve  upon  it  becomes  a  line  of  curvature  on  the  deform. 

This  result  suggests  the  following  problem  of  Bonnet*: 
To  determine  the  surfaces  which  can  be  deformed  with  preservation 
of  their  lines  of  curvature. 

*  Memoire  sur  la  theorie  des  surfaces  applicables  sur  une  surface  donnee,  Journal  de 
I'Ecole  Poly  technique,  Cahier  42  (1867),  p.  58. 


LINES  OF  CUBVATUBE  IN  COBBESPONDENCE      337 

We  follow  the  method  of  Bonnet  in  making  use  of  the  funda 
mental  equations  in  the  form  (V,  48,  55).  We  assume  that  the  lines 
of  curvature  are  parametric.  In  this  case  these  equations  reduce  to 

($>i     _      dct-~~      Sr  ^ 

(24) 


dv 

From  these  equations  it  follows  that  if  $  and  S'  are  two  applicable 
surfaces  referred  to  corresponding  lines  of  curvature,  the  functions 
r  and  rl  have  the  same  value  for  both  surfaces,  and  consequently 
the  same  is  true  of  the  product  qpr  Hence  our  problem  reduces 
to  the  determination  of  two  sets  of  functions  pv  q  ;  p[,  q',  satisfying 
the  above  equations.  In  consequence  of  the  identity 

(25)  p(q'=P& 

we  have  from  the  first  two  of  (24) 

*£-»£•    <S?-S- 

of  which  the  integrals  are  p'i2  =  p?+f(v),  q'2  =  <f+  <#>(M),  where  f(v) 
and  <f>(u)  are  functions  of  v  and  u  respectively.  The  parameters  w, 
v  may  be  chosen  so  that  these  functions  become  constants  #,  /3, 
and  consequently 

(27)  ;>I2  =  K+«»     ?'2=<Z2+£- 

If  these  equations  be  multiplied  together,  the  resulting  equation  is 
reducible  by  means  of  (25)  to  either  of  the  forms 

(28)  pi  ft  +  (fa  +  a(3  =  0,     p(*P  +  z'*a-a&  =  b. 

From  the  first  we  see  that  a  and  fi  cannot  both  be  positive  if  S  is  real, 
and  from  the  second  that  they  cannot  both  be  negative.  We  assume 
that  a  is  negative  and  j3  positive,  and  without  loss  of  generality  write 

(29)  rf-rf-li    ^-tf+l- 

The  first  of  (28)  reduces  to  pi—  q*  =  l.    In  conformity  with  this 
we  introduce  a  function  o>,  thus 

pl  =  cosh  &>,     q  =  sinh  o>. 
Then  equations  (29)  may  be  replaced  by 

p[  =  sinh  ft),     ^'  =  cosh  o>. 


338  DEFORMATION  OF  SURFACES 

Moreover,  the  fundamental  equations  (24)  reduce  to 


_ 


dv  du  V  ]£    du 

a  2  o2 

o>   ,   c  o>  .   ,  , 

—  :  H  --  =  —  smh  o>  cosh  o>. 
dt*a      0v- 

Comparing  these  results  with  §  118,  we  see  that  the  spherical 
representation  of  lines  of  curvature  of  the  surfaces  S  and  Sf  respec 
tively  is  the  same  as  of  the  lines  of  curvature  of  a  spherical  surface 
and  of  its  Hazzidakis  transform.  Conversely,  we  have  that  every 
surface  of  this  kind  admits  of  an  applicable  surface  with  lines  of 
curvature  in  correspondence. 

The  preceding  investigation  rested  on  the  hypothesis  that  neither 
the  first  nor  second  of  equations  (24)  vanishes  identically.  Suppose 
that  the  second  vanishes  ;  then  q  is  a  function  of  u  alone,  say  $(u). 
Since  the  product  p^q  differs  from  the  total  curvature  only  by  a 
factor  (cf.  §  70),  pl  cannot  be  zero  ;  therefore  r  =  0  and  q'  —  ^(u). 
Equation  (25)  is  now  of  the  form  pl<f>(u)=p[<t>l(u).  If  p[  be  elimi 
nated  from  this  equation  and  the  first  of  (27),  it  is  found  that  pl 
also  is  a  function  of  u  alone.  Hence  the  curves  v  =  const,  on  the 
sphere  are  great  circles  with  a  common  diameter,  and  therefore  S 
is  a  molding  surface  (§  130).  The  parameter  u  may  be  chosen  so 
that  we  may  take  q  =  1  and  p^=U\  then  from  (27)  and  (25)  we 
find  />(  =  Vf/2-h  #,  q'=  U/^/U*  +  a,  where  a  is  an  arbitrary  constant. 
Hence  we  have  the  theorem  : 

A  necessary  and  sufficient  condition  that  a  surface  admit  of  an 
applicable  surface  with  lines  of  curvature  in  correspondence  is  that 
the  surface  have  the  same  spherical  representation  of  its  lines  of  cur 
vature  as  a  spherical  surface  2,  or  be  a  molding  surface  ;  in  the  first 
case  there  is  one  applicable  surface,  and  the  spherical  representation 
of  its  lines  of  curvature  is  the  same  as  of  the  Hazzidakis  transform 
of  2  ;  in  the  second  case  there  is  an  infinity  of  applicable  surfaces.* 

141.  Conjugate  systems  in  correspondence.  When  two  surfaces 
are  applicable  to  one  another,  there  is  a  system  of  corresponding 
lines  which  is  conjugate  for  both  surfaces  (cf.  §  56).  The  results 
of  §  140  show  that  for  a  given  conjugate  system  on  a  surface  S 

*  Cf  .  EX.  14,  p.  319. 


CONJUGATE  SYSTEMS  IN  COEEESPONDENCE       339 

there  is  not  in  general  a  surface  Sl  applicable  to  S  with  the  corre 
sponding  system  conjugate.  We  inquire  under  what  conditions  a 
given  conjugate  system  of  S  possesses  this  property. 

Let  S  be  referred  to  the  given  conjugate  system.    If  the  corre 
sponding  system  on  an  applicable  surface  Sl  is  conjugate,  we  have 

D'  =  D[=  0,          Dip?  =  DD"  ; 

for  the  total  curvature  of  the  two  surfaces  is  the  same.  We  replace 
this  equation  by  the  two 

D!  -  tanh  6  •  D,          D['  =  coth  6  .  D", 
thus  defining  a  function  6.    The  Codazzi  equations  for  S  are 


Since  these  equations  must  be  satisfied  by  Dl  and  D",  we  have 
30  22     D  ,  c6          fll    Z 


lz' 
The  condition  of  integrability  of  (30)  is  reducible  to 


2    D 

m 


As  the  two  roots  of  this  equation  differ  only  in  sign,  and  thus  lead 
to  symmetric  surfaces,  we  need  consider  only  one.  If  it  be  substi 
tuted  in  (30),  we  obtain  two  conditions  upon  E,  F,  G  ;  7),  D",  which 
are  necessary  in  order  that  S  admit  of  an  applicable  surface  of  the 
kind  sought.  Hence  in  general  there  is  no  solution  of  the  problem. 
However,  if  the  two  expressions  in  the  brackets  of  (31)  vanish 
identically,  the  conditions  of  integrability  of  equations  (30)  are 
completely  satisfied,  and  S  admits  of  an  infinity  of  applicable  sur 
faces  upon  which  the  coordinate  curves  form  a  conjugate  system. 
Consequently  we  have  the  theorem  : 

If  a  conjugate  system  on  a  surface  S  corresponds  to  a  conjugate 
system  on  more  than  one  surface  applicable  to  S,  it  corresponds  to  a 
conjugate  system  on  an  infinity  of  surfaces  applicable  to  S. 


340  DEFORMATION  OF  SURFACES 

We  shall  give  this  result  another  interpretation  by  considering 
the  spherical  representation  of  S.  From  (VI,  38)  we  have 

1-22-1   .D  f!2V  firi-D".      /12V 

ii/^=  •la/'      la/3"  ~li/' 

{rtV 
>  are  formed  with  respect  to  the  linear  ele 

ment  of  the  spherical  representation  of  S.    If  we  substitute  these 
values  in  (30),  we  get 

d6      f!2\'  d6     /12V      ... 

—  =  s        f  tanh  0,         —  =  \       \  coth  0, 

a^    1  2  J  0*    1  1  J 

and  the  condition  that  these  equations  have  an  integral  involving 
a  parameter  becomes 

a  ri2V_  a  ri2V     py  ri2V 
Sii/~Sla./     "-iJisr 

The  first  of  these  equations  is  the  condition  that  the  curves 
upon  the  sphere  represent  the  asymptotic  lines  upon  a  certain  sur 
face  2  (cf.  §  78).  Moreover,  if  K  denotes  the  total  curvature  of 
S,  and  we  put  K=  —  l//>2»  we  have 


(34) 

du  2 

Now  equations  (33)  are  equivalent  to  (34),  and 
&  log  p 


r,<2 

which  reduces  to  —  —  =  0.   As  the  general  integral  of  this  equation 
cucv 

is  p  =  cf)(u)  4-  ^(i'),  where  (/>  and  ^  are  arbitrary  functions  of  u  and 
v  respectively,  we  have  the  following  theorem  due  to  Bianchi*  : 

A  necessary  and  sufficient  condition  that  a  surface  S  admit  a  con 
tinuous  deformation  in  which  a  conjugate  system  remains  conjugate 
is  that  the  spherical  representation  of  this  system  be  that  of  the  asymp 
totic  lines  of  a  surface  whose  total  curvature,  expressed  in  terms  of 
parameters  referring  to  these  lines,  is  of  the  form 


*  Annali,  Ser.  2,  Vol.  XVIII  (1890),  p.  320;  also  Lezioni,  Vol.  II,  p.  83. 


CONJUGATE  SYSTEMS  IN  COKKESPONDENCE       341 

The  pseudospherical  surfaces  afford  an  example  of  surfaces  with 
K  of  this  form.  In  this  case  </>  and  ^r  are  constants,  so  that  equa- 

("12")  '     f  12"|  ' 
tions  (34)  reduce  to  1       \  ~\  a  |  •**•  ^'  which,  in  consequence  of 

{11"|      f22>| 
f  =  1  ^,    f=0.    But  these  are  the  condi 

tions  that  the  parametric  curves  on  S  be  geodesies.  A  surface  with 
a  conjugate  system  of  geodesies  is  called  a  surface  of  l/ross.  We 
state  these  results  thus  : 

A  surface  of  l^oss  admits  of  a  continuous  deformation  in  which  the 
geodesic  conjugate  system  is  preserved  ;  consequently  all  the  new  sur 
faces  are  of  the  same  kind. 

EXAMPLES 

1.  Show  that  every  integral  of  the  equation  Ai0  =  1  is  an  integral  of  the  funda 
mental  equation  (18). 

2.  On  a  right  helicoid  the  helices  are  asymptotic  lines.    Find  the  surfaces  appli 
cable  to  the  helicoid  in  such  a  way  that  one  of  the  helices  is  unaltered  in  form  and 
continues  to  be  an  asymptotic  line. 

3.  A  surface  applicable  to  a  surface  of  revolution  with  the  lines  of  curvature 
on  the  two  surfaces  in  correspondence  is  a  surface  of  revolution. 

4.  Show  that  the  equations 

X=KTCOS-,        y  =  train-,        z  =  /Vl  —  /c2r'2c?M, 

K  K  J 

define  a  family  of  applicable  surfaces  of  revolution  with  lines  of  curvature  in  corre 
spondence.  Discuss  the  effect  of  a  variation  of  the  parameter  K. 

5.  Let  S  denote  a  surface  parallel  to  a  spherical  surface  S.    Find  the  surface 
applicable  to  S  with  preservation  of  the  lines  of  curvature. 

6.  It  Si  and  S2  be  applicable  surfaces  referred  to  the  common  conjugate  sys 
tem,  their  coordinates  &i,  y\,z\\  £2,  ?/2,  £2  are  solutions  of  the  same  point  equation 
(cf.  VI,  26),  and  the  function  xf  -f  y?  +  zf  —  (x|  +  y.|  +  z|)  also  is  a  solution. 

7.  Show  that  the  locus  of  a  point  which  divides  in  constant  ratio  the  join  of 
corresponding  points  on  the  surfaces  Si  and  <S2  of  Ex.  6  is  a  surface  upon  which  the 
parametric  lines  form  a  conjugate  system.    Under  what  condition  is  this  surface 
applicable  to  Si  and  /S2  ? 

8.  The  tetrahedral  surface 

x  =  A(a  +  u)*(a  +  «)*,     y  =  B(b  +  u)*(b  +  v)*,     z  =  C(c  +  w)*(c  +  v)f, 
admits  of  an  infinity  of  deforms 


The  curves  u  =  v  upon  these  surfaces  are  congruent,  and  consequently  each  is  an 
asymptotic  line  on  the  surface  through  it. 


342  DEFORMATION  OF  SURFACES 

9.  If  the  equations  of  a  surface  are  of  the  form 

x  =  U1Vll        y  =  UiV!,        z=V*t 
the  equations 


sin  0, 


where  h  denotes  a  constant,  define  a  family  of  applicable  surfaces  upon  which  the 
parametric  lines  form  a  conjugate  system. 

10.  Show  that  the  equations  of  the  quadrics  can  be  put  in  the  form  of  Ex.  9, 
and  apply  the  results  to  this  case. 

142.  Asymptotic  lines  in  correspondence.  Deformation  of  a  ruled 
surface.  We  have  seen  (§  139)  that  a  surface  can  be  subjected  to 
a  continuous  deformation  in  which  an  asymptotic  line  remains 
asymptotic.  We  ask  whether  two  surfaces  are  applicable  with 
the  asymptotic  lines  in  one  system  corresponding  to  asymptotic 
lines  of  the  other.  We  assume  that  there  are  two  such  surfaces,  S, 
Slt  and  we  take  the  corresponding  asymptotic  lines  for  the  curves 
v  =  const,  and  their  orthogonal  trajectories  for  u  =  const.  In  con 
sequence  of  this  choice  and  the  fact  that  the  total  curvature  of  the 
two  surfaces  is  the  same,  we  have 

(36)  J9  =  D1=0,         JF=0,         D'  =  D[. 

The  Codazzi  equations  (V,  13')  for  S  reduce  to 

< 


Q 

Because  of  (36)  the  Codazzi  equation  for  S1  analogous  to  the  first 
of  (37)  will  differ  from  the  latter  only  in  the  last  term.  Hence  we 
must  have  either  D"  =  Z>",  or  E—f(u}.  In  the  former  case  the  sur 
faces  S  and  Sl  are  congruent.  Hence  we  are  brought  to  the  second, 
which  is  the  condition  that  the  curves  v  =  const,  be  geodesies.  As 
the  latter  are  asymptotic  lines  also,  they  are  straight,  and  conse 
quently  8  must  be  a  ruled  surface.  By  changing  the  parameter  w, 
we  have  J5?  =  l,  and  equations  (37)  reduce  to 


EULED  SURFACES  343 

By  a  suitable  choice  of  the  parameter  v  the  first  of  these  equations 
may  be  replaced  by  JX=1/V5,  and  the  second  becomes 


ra/i 

=  I  ~(- 

J  9*\& 


where  </>  is  an  arbitrary  function.  These  results  establish  the  fol 
lowing  theorem  of  Bonnet  : 

A  necessary  and  sufficient  condition  that  a  surface  admit  an 
applicable  surface  with  the  asymptotic  lines  in  one  system  on  each 
surface  corresponding  is  that  the  surface  be  ruled;  moreover,  a 
ruled  surface  admits  of  a  continuous  deformation  in  which  the 
generators  remain  straight. 

To  this  may  be  added  the  theorem  : 

If  two  surfaces  are  applicable  and  the  asymptotic  lines  in  both 
systems  on  each  surface  are  in  correspondence,  the  surfaces  are  con 
gruent,  or  symmetric. 

This  is  readily  proved  when  the  asymptotic  lines  are  taken  as 
parametric. 

We  shall  establish  the  second  part  of  the  above  theorem  in 
another  manner.  For  this  purpose  we  take  the  equations  of  the 
ruled  surface  in  the  form  (§  103) 

(38)  x  =  xQ+lu,         y  =  yQ+mu,          z  =  z0+nu, 

where  XQ,  y^,  z0  are  the  coordinates  of  the  directrix  C  expressed  as 
functions  of  its  arc  v,  and  I,  m,  n  are  the  direction-cosines  of  the 
generators,  also  functions  of  v.  They  satisfy  the  conditions 

(39)  aJ'+jtf-K1-!.         *2+wa+na  =  l, 

where  the  accents  indicate  differentiation  with  respect  to  v. 
Furthermore,  the  linear  element  is 

(40)  ds*=  du2+  2  cos  00dudv  +  (aV+  2 

'-2     n'       b  =  l'x' 


Hence  if  we  have  a  ruled  surface  with  the  linear  element  (40),  the 
problem  of  finding  a  ruled  surface  applicable  to  it,  with  the  gener 
ators  of  the  two  surfaces  corresponding,  reduces  to  the  determi 
nation  of  six  functions  of  v,  namely  XQ,  y0,  z0;  I,  m,  n,  satisfying 


344  DEFORMATION  OF  SURFACES 

the  five  conditions  (39),  (41).  From  this  it  follows  that  there  is  an 
arbitrary  function  of  v  involved  in  the  problem,  and  consequently 
there  is  an  infinity  of  ruled  surfaces  with  the  linear  element  (40). 

There  are  two  general  ways  in  which  the  choice  of  this  arbi 
trary  function  may  be  made,  —  either  as  determining  the  form  of 
the  director-cone  of  the  required  surface,  or  by  a  property  of  the 
directrix.  We  consider  these  two  cases. 

143.  Method  of  Minding.  The  first  case  was  studied  by  Mind 
ing.*  He  took  /,  m,  n  in  the  form 

(42)  /  =  cos  <£  cos  i/r,        m  =  cos  <£  sin  i/r,       n  =  sin  </>, 

which  evidently  satisfy  the  second  of  (39).  The  first  of  (41) 
reduces  to 

(43)  <J>'2+^'2cos2£=aa. 

If  we  solve  equations  (39)  and  (41)  for  x'^  y'Q,  z0',  the  resulting 
expressions  are  reducible  by  means  of  (VII,  63)  to 


(44) 


'Q  =  I  cos  6Q  +  ^  [I'b  ±  (mnf-  m'n)  VV  sin200-  62], 


and  analogous  expressions  for  y[  and  z[.  Hence,  if  <£  be  an  arbi 
trary  function  of  v,  and  ^  be  given  by 

(46)  +=(^f^ 

J        COS  (/> 

the  functions  #0,  #0,  z0,  obtained  from  (44)  by  quadratures,  together 
with  /,  m,  n  from  (42),  determine  a  ruled  surface  with  the  linear 
element  (40). 

Each  choice  of  (/>  gives  a  different  director-cone,  which  is  deter 
mined  by  the  curve  in  which  the  cone  cuts  the  unit  sphere,  whose 
center  is  at  the  vertex  of  the  cone.  Such  a  curve  is  defined  by  a 
relation  /(c/>,  -<fr)  =  0,  so  that  instead  of  choosing  $  arbitrarily  we 
may  take  /  as  arbitrary;  for,  by  combining  equations  (43)  and 
/(<£,  ifr)  —  0,  we  obtain  the  expressions  for  <£  and  ^r  as  functions 
of  v.  Hence  : 

A  ruled  surface  may  be  deformed  in  such  a  way  that  the  director- 
cone  takes  an  arbitrary  form. 

*  Crelle,  Vol.  XVIII  (1838);  pp.  297-302. 


RULED  SURFACES  345 

When  the  given  ruled  surface  is  nondevelopable,  the  radicand 
in  (44)  is  different  from  zero,  and  consequently  there  are  two  dif 
ferent  sets  of  functions  XQJ  yQ1  ZQ.  Hence  there  are  two  applicable 
ruled  surfaces  with  the  same  director-cone.  If  the  parameters  of 
distribution  of  these  two  surfaces  be  calculated  by  (VII,  73),  they 
are  found  to  differ  only  in  sign.  Hence  we  have  the  theorem  of 
Beltrami :  * 

A  ruled  surface  admits  of  an  applicable  ruled  surface  such  that 
corresponding  generators  are  parallel,  and  the  parameters  of  distri 
bution  differ  only  in  sign. 

144.  Particular  deformations  of  ruled  surfaces.  By  means  of  the 
preceding  results  we  prove  the  theorem : 

A  ruled  surface  may  be  deformed  in  an  infinity  of  ways  so  that  a 
given  curve  becomes  plane. 

Let  the  given  curve  be  taken  for  the  directrix  of  the  original 
surface.  Assuming  that  a  deform  of  the  kind  desired  exists,  we 
take  its  plane  for  the  zy-plane.  From  (44)  we  have 


a2n  cos  00  +  bnf  ±  (lmf  —  I'm)     a2  sin2  00  —  62  =  0, 
which,  in  consequence  of  (42)  arid  (43),  reduces  to 


b  cosc/>.(£'+  a2  sine/)  cos#0±  cosc^Va2  —  <//2Va2  sin200—  b2=  0. 

The  integral  of  this  equation  involves  an  arbitrary  constant,  and 
thus  the  theorem  is  proved. 

The  preceding  example  belongs  to  the  class  of  problems  whose 
general  statement  is  as  follows: 

To  deform  a  ruled  surface  into  a  ruled  surface  in  such  a  way  that 
the  deform  of  a  given  curve  C  on  the  original  surface  shall  possess  a 
certain  property  on  the  resulting  surface. 

We  consider  this  general  problem.  Let  the  deform  of  C  be  the 
directrix  of  the  required  surface,  and  let  «0,  /30,  70;  J0,  TWO,  n0;  X0,  /*0,  v0 
denote  the  direction-cosines  of  its  tangent,  principal  normal,  and 
binormal.  If  <r  denotes  the  angle  between  the  osculating  plane  to 
the  curve  and  the  tangent  plane  to  the  surface,  we  have 

(46)  I  =  #0  cos  #0  -f  sin  00  (/0  cos  <r  +  X0  sin  <r), 

*  Annali,  Vol.  VII  (1865),  p.  115. 


346  DEFORMATION  OF  SURFACES 

and  similar  expressions  for  m  and  n.    When  these  values  are  sub 
stituted  in  the  first  two  of  equations  (41),  the  resulting  equations 
are  reducible,  by  means  of  the  Frenet  formulas  (I,  50),  to 
coscr          /-,  ,       b 


(47) 


P 
~cos0n  „  „,,  .  sin  cr  sin 


.    n   .      sincr  sin#0~]2 
(cos  cr  sin  00)'  H 

T/  •      aM       COS  cr  sin  ft"] 2          „        I2 

+    (sin  cr  sin  0Q) ' \—a—b. 


These  are  two  equations  of  condition  on  cr,  /o,  T,  as  functions  of  v. 
Each  set  of  solutions  determines  a  solution  of  the  problem ;  for, 
the  directrix  is  determined  by  expressions  for  p  and  r,  and  equa 
tions  (46)  give  the  direction-cosines  of  the  generators. 

We  leave  it  to  the  reader  to  prove  the  above  theorem  by  this 
means,  and  we  proceed  to  the  proof  of  the  theorem: 

A  ruled  surface  may  be  deformed  in  such  a  manner  that  a  given 
curve  C  becomes  an  asymptotic  line  on  the  new  ruled  surface. 

On  the  deform  we  must  have  a  =  0  or  a  =  TT,  so  that  from  (47) 


p 

the  sign  being  fixed  by  the  fact  that  p  is  necessarily  positive.  The 
second  of  (47)  reduces  to 


sin26> 


If  the  curve  with  these  intrinsic  equations  be  constructed,  and  in 
the  osculating  plane  at  each  point  the  line  be  drawn  which  makes 
the  angle  00  with  the  tangent,  the  locus  of  these  lines  is  a  ruled 
surface  satisfying  the  given  conditions. 

When  the  curve  C  is  an  orthogonal  trajectory  of  the  generators, 
the  same  is  true  of  its  deform.  Hence : 

A  ruled  surface  may  be  deformed  in  such  a  way  that  all  the  gener 
ators  become  the  principal  normals  of  the  deform  of  any  one  of  their 
orthogonal  trajectories. 

Having  thus  considered  the  deformation  of  ruled  surfaces  in 
which  the  generators  remain  straight,  we  inquire  whether  two 


RULED  SURFACES  347 

ruled  surfaces  are  applicable  with  the  generators  of  each  corre 
sponding  to  curves  on  the  other.  Assume  that  it  is  possible,  and 
let  v  =  const,  be  the  generators  of  S  and  u  =  const,  the  curves 
on  S  corresponding  to  the  generators  of  Sr  From  (V,  13)  it 
follows  that  the  conditions  for  this  are  respectively 


where  K=  —  \/p\  But  equations  (48)  are  the  necessary  and 
sufficient  conditions  that  there  be  a  surface  2  applicable  to  S 
and  Sv  upon  which  the  asymptotic  lines  are  parametric  (cf.  VI,  3). 
But  the  curves  v  =  const,  and  u  =  const,  are  geodesies  on  S  and  8^ 
and  consequently  on  2.  Therefore  2  is  doubly  ruled.  Hence  : 

If  two  ruled  surfaces  S  and  Sl  are  applicable  to  one  another,  the 
generators  correspond  unless  the  surfaces  are  applicable  to  a  quadric 
with  the  generators  of  S  and  Sl  corresponding  to  the  two  different 
systems  of  generators  of  the  quadric. 

EXAMPLES 

1.  A  ruled  surface  can  be  deformed  into  another  ruled  surface  in  such  a  way 
that  a  geodesic  becomes  a  straight  line. 

2.  A  ruled  surface  formed  by  the  binomials  of  a  curve  C  can  be  deformed  into 
a  right  conoid  ;  the  latter  is  the  right  helicoid  when  the  torsion  of  C  is  constant. 
Prove  the  converse  also. 

3.  On  the  hyperboloid  of  revolution,  defined  by 

xwu.v          y      u   .    v  v          z      u 

-  =  —  cos  -  +  sin  -  ,         -  =  —  sin  --  cos  -  ,         -  =  —  . 
c       A        c  c  c       A        c  c          d      A 

where  A2  =  c2  +  d2,  the  circle  of  gorge  is  a  geodesic,  which  is  met  by  the  generators 
under  the  anle  cos- 


4.  Show  that  the  ruled  surface  which  results  from  the  deformation  of  the 
hyperboloid  of  Ex.  3,  in  which  the  circle  of  gorge  becomes  straight,  is  given  by 

ud        u  ud   .    v  uc 

x  =  —  cos  -  ,         y  —  —  sin  -  ,         z  =  --  \-  v. 

Ad  Ad  A 

5.  Show  that  the  ruled  surface  to  which  the  hyperboloid  of  Ex.  3  is  applicable 
with  parallelism  of  corresponding  generators  is  the  helicoid 

x      u       v      c2  —  d2   .    v        y      u   .    v      c2  —  d2        v        z      u      2  c 

-  =  —  cos  -  H  --  sin  -i       -  =  —  sin  -  ---  cos  -  ,       -  =  —  I  --  -  v, 
c      A        c      C2  +  d2       c         c      A        c      c2  +  d2        c        d      A      A2    ' 

and  that  the  circle  of  gorge  of  the  former  corresponds  to  a  helix  upon  the  latter. 

6.  When  the  directrix  is  a  geodesic,  equations  (47)  reduce  to 

Bin  *0.*;  +  6  =  0, 


348  DEFORMATION  OF  SURFACES 

7.  When  an  hyperboloid  of  revolution  of  one  sheet  is  deformed  into  another 
ruled  surface,  the  circle  of  gorge  becomes  a  Bertraml  curve  and  the  generators 
are  parallel  to  the  corresponding  bmormals  of  the  conjugate  Bertrand  curve. 

8.  A  ruled  surface  can  be  deformed  in  such  a  way  that  a  given  curve  is  made 
to  lie  upon  a  sphere  of  arbitrary  radius. 

9.  When  a  ruled  surface  admits  a  continuous  deformation  into  itself  the  total 
curvature  of  the  surface  is  constant  along  the  line  of  striction,  the  generators  meet  the 
latter  under  constant  angle,  and  the  parameter  of  distribution  is  constant  (cf.  §  126). 

10.  Two  applicable  ruled  surfaces  whose  corresponding  generators  are  parallel 
cannot  be  obtained  from  one  another  by  a  continuous  deformation. 

GENERAL  EXAMPLES 

1.  Determine  the  systems  of  coordinate  lines  in  the  plane  such  that  the  linear 
element  of  the  plane  is  ^U2  _j_  ^2 

=  ' 


where  U  and  V  are  functions  of  u  and  u  respectively. 

2.  Solve  for  the  sphere  the  problem  similar  to  Ex.  1. 

3.  Determine  the  functions  0(w)  and  ^  (u)  so  that  the  helicoids,  defined  by 


x  =  a\/U2-  6'2cos-,     y  = 


shall  be  applicable  to  the  surface  whose  equations  are 


where  U  is  any  function  of  u. 

4.  Apply  the  method  of  Ex.  3  to  find  helicoids  applicable  to  the  pseudosphere  ; 
to  the  catenoid. 

5.  The  equations 

x  =  a  V2  u  —  2  cos  - ,     y  =  a  V'2  it  —  2  sin  -- ,     z  =  -  (u  —  1) 
a  a  2 

define  a  paraboloid  of  revolution.  Show  that  surfaces  applicable  to  it  are  defined  by 

id  r  /*  /* 

X  -  —     /302  ~/203  +  J    (fzdfz  -fsdfz)  -  J   (02^03  ~  03  dfa)     , 

»-? 

2  =  -—I  /201  —  /102 


where  a  is  a  real  constant,  and  the  /'s  and  0's  are  functions  of  a  parameter  a  and 
/3  respectively  such  that 


6.  Investigate  the  special  case  of  Ex.  5  for  which  a  and  /3  are  conjugate  imaginary 
functions,  and  2  +  a-2a*  .2-a-2a* 

fl  =  —      -  7=-  -  '      /2  =  l  —  ,  -  -  '      /•  =  *« 

2V2a  2V2a 

and  the  0's  are  functions  conjugate  imaginary  to  the/'s. 


GENERAL  EXAMPLES  349 

7.  Show  that  the  surface  of  translation 

x—  a(cosw  +  cosv),     y  —  a(sinw  +  sinv),     z  =  c(u  +  v) 
is  applicable  to  a  surface  of  revolution. 

8.  Show  that  the  minimal  surfaces  applicable  to  a  spiral  surface  (Ex.  22,  p.  151) 
are  determined  by  the  functions  F(u)  =  cum  +  in,  4>(u)  =  ciom~in,  and  that  the  asso 
ciate  surfaces  are  similar  to  the  given  one. 

9.  If  the  coefficients  E,  F,  6?  of  the  linear  element  of  a  surface  are  homogeneous 
functions  of  u  and  v  of  order  —  2,  the  surface  is  applicable  to  a  surface  of  revolution. 

10.  If  z,  y,  z  are  the  coordinates  of  a  surface  S  referred  to  a  conjugate  system, 
the  equations 

ctf__      dx     W__pdy_    ^i_p^:.     ^L-Q^L    ^-Q^y.    —  =  Q~ 

aw  ~    aw'  aw  ~    au'  aw~    aw'    au       au'  au  ~    aw'  cv  ~    au 

are  integrable  if  P  and  Q  satisfy  the  conditions 


where  the  Christoffel  symbols  are  formed  with  respect  to  the  linear  element  of  S. 
Show  that  on  the  surface  S',  whose  coordinates  are  x',  y'  ',  z',  the  parametric  curves 
form  a  conjugate  system,  and  that  the  normals  to  S  and  S'  at  corresponding  points 
are  parallel. 

11.  Show  that  for  the  surface 

x  -  f\fi(u)du  +  0i(w),     y  =  f  A/2(w)dit  +  02(u),     z  =  j'\f3(u)du  +  03(u), 

where  \  is  any  function  of  u  and  u,  and/!,  /2,  /3  ;  0i,  02,  03*are  functions  of  u  and 
u  respectively,  the  parametric  curves  form  a  conjugate  system.  Apply  the  results 
of  Ex.  10  to  this  surface,  and  discuss  the  case  for  which  X  is  independent  of  v. 

12.  If  S  and  Si  are  two  applicable  surfaces,  and  S{  denotes  the  surface  corre 
sponding  to  Si  in  the  same  manner  as  S'  to  S  in  Ex.  10  and  by  means  of  the  same 
functions  P  and  Q,  then  S'  and  S{  are  applicable  surfaces. 

13.  If  x,  ?/,  z  and  «i,  2/1,  z\  are  the  coordinates  of  a  pair  of  applicable  surfaces 
S  and  Si,  a  second  pair  of  applicable  surfaces  S'  and  S{  is  denned  by 

x'  =  x  +  h(z  +  zi)  -  k(y  +  T/J),        x[  =  xl-  h(z  +  zi)  +  k(y  +  2/1), 

y'  =  y  +  k(x  +  xi)-g(z  +  *i),        yi  =  z/i  -  k  (x  +  xi)  +  g(z  +  zi), 

z'  =  z  +  g(y  +  yi)  -  h(x  +  xx),        zi  =  zt  -  g  (y  +  z/i)  +  h(x  +  KI), 

where  #,  ^-,  and  fc  are  constants.    Show  that  the  line  segments  joining  correspond 

ing  points  of  S  and  S'  are  equal  and  parallel  to  those  for  Si  and  S{  ;  that  the  lines 

joining  corresponding  points  on  S  and  Si  meet  the  similar  lines  for  S'  and  S{  ;  and 

that  the  common  conjugate  system  on  S  and  Si  corresponds  to  the  common  conju 

gate  system  on  S'  and  Si. 

14.  Apply  the  results  of  Ex.  13  to  the  surfaces  of  translation 

x  =  w2  -  v2  -|-  2  av,     y  =  2  w2  -I-  v2  -  2  au  -  2*  V&2  +  3  w2dw,     z  =  2  6u, 


2  u2  -  2  au  -  2 

Z!  =  2  fa2  -  3u2dv. 
Show  that  when  g  =  h  =  0,  k  =  -  1,  the  surface  S'  is  an  elliptic  paraboloid. 


350  DEFORMATION  OF  SURFACES 

15.  Show  that  the  equations 

'"a2"    "'     y '  ~  J 

where  the  accent  indicates  differentiation  with  respect  to  the  argument,  define  a 
family  of  applicable  surfaces  of  translation.   Apply  the  results  of  Ex.  12  to  this  case. 

16.  Show  that  when  S  and  Si  in  Exs.  12  and  13  are  surfaces  of  translation,  and 
their  generating  curves  correspond,  the  same  is  true  of  S/  and  S{. 

17.  If  lines  be  drawn  through  points  of  a  Bertrand  curve  parallel  to  the  binor- 
mals  of  the  conjugate  curve,  their  locus  is  applicable  to  a  surface  of  revolution. 

18.  If  a  real  ruled  surface  is  applicable  to  a  surface  of  revolution,  it  is  applicable 
to  the  right  helicoid  or  to  a  hyperboloid  of  revolution  of  one  sheet  (cf.  Ex.  9,  §  144). 

19.  A  ruled  surface  can  be  deformed  in  an  infinity  of  ways  so  that  a  curve  not 
orthogonal  to  the  generators  shall  be  a  line  of  curvature  on  the  new  ruled  surface, 
unless  the  given  curve  is  a  geodesic ;  in  the  latter  case  the  deformation  is  unique 
and  the  line  of  curvature  is  plane. 

20.  Let  P  be  any  point  of  a  twisted  curve  C,  and  MI,  M2  points  on  the  principal 

normal  to  C  such  that  /  /»  /f « 

=  -  PM2  =  a  sin  (  {  —  H 


where  a,  6  are  constants  and  p  is  the  radius  of  curvature  of  C.    The  loci  of  the  lines 
through  M\  and  3f2  parallel  to  the  tangent  to  C  at  P  are  applicable  ruled  surfaces. 

21.  On  the  surface  whose  equations  are 

x  =  M,     y  =f(u)<f>'(v)  +  i//(v),     z  =  /(u)[0(i>)  -  00'(u)]  +  t(o)-  fl^'(u), 
the  parametric  curves  form  a  conjugate  system,  the  curves  u  =  const,  lie  in  planes 
parallel  to  the  yz-plane,  and  the  curves  v  —  const,  in  planes  parallel  to  the  x-axis  ; 
hence  the  tangents  to  the  curves  u  =•  const,  at  their  points  of  intersection  with  a 
curve  v  =  const,  are  parallel. 

22.  Investigate  the  character  of  the  surfaces  of  Ex.  21  in  the  following  cases  : 
(a),  0  (v)  =  Vv2  +  l  ;  (b),  0  (u)  =  const.  ;  (c),  t(v)  =  Q;  (d),  f(u)  =  au  +  b. 

23.  If  the  equations  of  Ex.  21  be  written 


the  most  general  applicable  surfaces  of  the  same  kind  with  parametric  curves  cor 
responding  are  defined  by 


where  AC  is  a  parameter,  and  the  functions  4>i,  $2,  ^i,  ^2  satisfy  the  conditions 

$2  +  <I>|  =  02  +  0|  -  K,  4>i*  +  <J>22  =  0{2  +  0£2, 

~  $1(0212 


Show  also  that  the  determination  of  4>i  and  3>2  requires  only  a  quadrature. 


CHAPTER  X 

DEFORMATION  OF  SURFACES.    THE  METHOD  OF  WEINGARTEN 

145.  Reduced  form  of  the  linear  element.  Weingarten  has  re 
marked  that  when  we  reduce  the  determination  of  all  surfaces  appli 
cable  to  a  given  one  to  the  solution  of  the  equation  (IX,  18),  namely 

(1)  ±J  =  (\-\6)K, 

we  make  no  use  of  our  knowledge  of  the  given  surface,  and  in 
reality  are  trying  to  solve  the  problem  of  finding  all  the  surfaces 
with  an  assigned  linear  element.  In  his  celebrated  memoir,  Sur  la 
deformation  des  surfaces,*  which  was  awarded  the  grand  prize  of 
the  French  Academy  in  1894,  Weingarten  showed  that  by  taking 
account  of  the  given  surface  the  above  equation  can  be  replaced 
by  another  which  can  be  solved  in  several  important  cases.  This 
chapter  is  devoted  to  the  exposition  of  this  method.  We  begin  by 
determining  a  particular  moving  trihedral  for  the  given  surface. 

It  follows  from  (VII,  64)  that  the  necessary  and  sufficient  con 
dition  that  the  directrix  of  a  ruled  surface  be  the  line  of  striction  is 

(2)  6  =  a#'+#X+*X=0. 

The  functions  //  m/  n'  are  proportional  to  the  direction-cosines  of 
the  curve  in  which  the  director-cone  of  the  surface  meets  the  unit 
sphere  with  center  at  the  vertex  of  the  cone.  We  call  this  curve 
the  spherical  indicatrix  of  the  surface.  From  (2)  and  the  identity 

ll'+mm'+  nn'=  0 

it  is  seen  that  the  tangent  to  the  spherical  indicatrix  is  perpen 
dicular  to  the  tangent  plane  to  the  surface  at  the  corresponding 
point  of  the  line  of  striction.  This  fact  is  going  to  enable  us 
to  determine  under  what  conditions  a  ruled  surface  2,  tangent 
to  a  curved  surface  S  along  a  curve  C,  admits  the  latter  for 
its  line  of  striction. 

*Acta  Mathematica,  Vol.  XX  (1896),  pp.  159-200. 
351 


352  DEFORMATION  OF  SURFACES 

We  suppose  that  the  parameters  w,  v  are  any  whatever,  and  that 
the  surface  is  referred  to  a  moving  trihedral.  We  consider  the 
ruled  surface  formed  by  the  z-axis  of  the  trihedral  as  the  origin 
of  the  latter  describes  the  curve  C.  The  point  (1,  0,  0)  of  a  second 
trihedral  parallel  to  this  one,  but  with  origin  fixed,  describes  the 
spherical  indicatrix  of  2.  From  equations  (V,  51)  we  find  that  the 
components  of  a  displacement  of  this  point  are 

0,          r  du  4-  r^v,          —  (qdu  +  q^dv). 

In  order  that  the  displacement  be  perpendicular  to  the  tangent 
plane  to  2  at  the  corresponding  point  of  (7,  that  is,  perpendicular 
to  the  zy-plane  of  the  moving  trihedral,  we  must  have  ' 

(3)  rdu  +  rldv  =  Q. 

Hence  if  a  trihedral  T  be  associated  with  a  surface  S  in  any  man 
ner,  as  the  vertex  of  T  describes  an  integral  curve  of  equation  (3), 
the  2>axis  of  T  generates  a  ruled  surface  whose  line  of  striction  is 
this  curve. 

When  the  parametric  lines  on  S  are  given,  and  also  the  angle  U 
which  the  a>axis  of  T  makes  with  the  tangent  to  the  curve  v  —  const., 
the  functions  r  and  rl  are  completely  determined,  as  follows  from 
(V,  52,  55).  They  are 

Hll         cU 


Hence  if  U  be  given  the  value 

///  C121 
It")  *+*<»>• 

where  <f>  (u)  denotes  an  arbitrary  function  of  M,  the  function  rt  is  zero, 
and  as  the  vertex  of  the  trihedral  describes  a  curve  u  =  const.,  the 
z-axis  describes  a  ruled  surface  whose  line  of  striction  is  this  curve. 
Suppose  now  that  the  trihedral  is  such  that  rx=  0.  From  (V,  48,  64) 
it  follows  that 

(6) 

consequently 

(1)  r=  C 

where  ty  is  an  arbitrary  function  of  u. 


PARTICULAR  TRIHEDRAL 


353 


Let  the  right-hand  member  of  (7)  be  denoted  by  f(u,  v),  and  change 
the  parameters  of  the  surface  in  accordance  with  the  equations 

ul=u,    v^  =  f(u,  v). 
From  §  32  and  equation  (7)  it  follows  that 


i- 

dv 

Since  K  is  unaltered  by  the  transformation,  in  terms  of  the  new 
coordinates  HVK  is  equal  to  unity,  and  hence  from  (6)  we  have 
r  =  vr    Therefore  the  coordinate  curves  and  the  moving  trihedral 
of  a  surface  can  be  chosen  in  such  a  way  tnat 
(8)  ^=0,          r  =  v,         HK=l. 

In  this  case  we  say  that  the  linear  element  of  the  surface  is  in  its 
reduced  form.  It  should  be  remarked  that  for  surfaces  of  negative 
curvature  the  parameters  are  imaginary. 

146.  General  formulas.  If  X^  Y^  Z^  ;  A;,  F2,  Zj>  X,  r,  Z  denote 
the  direction-cosines  of  the  axes  of  the  moving  trihedral  with 
respect  to  fixed  axes,  we  have,  from  (V,  47), 


(9) 


du 


i±l  —  _  Xa  — - 

dv   "  qi'  dv 


- 


dv 


The  rotations  p,p^  q<>  $i  satisfy  equations  (V,  48)  in  the  reduced  form 


dv       du 


dv       du 


The  coordinates  x,  y,  z  of  S  with  reference  to  these  fixed  axes  are 
given  by  c          /• 


(11) 


where 


y  = 


2  =  f(^i  +  ^2)  ^  +  (f  1^1  +  ^2)  ^N 


and 
(13) 


dv      du  '" 


354  DEFORMATION  OF  SURFACES 

Weingarten's  method  consists  in  replacing  the  coefficients  of 
f*  77,  ft,  rjl  in  the  last  of  equations  (13)  by  differential  parameters 
of  u  formed  with  respect  to  the  linear  element  of  the  spherical 
representation  of  the  z-axis  of  the  moving  trihedral. 

By  means  of  (9)  this  linear  element  is  reducible  to 

(14)  da2  =  dX2  +  dY2  +  dZ2  =  (v2  +  q2)  du2  +  2  qq^dudv  +  q2dv2. 

The   differential   parameters   of  u,   formed  with    respect    to    this 
form,  have  the  values  * 

I  _4(y2+22) 

(15)  ^  v*ti 

A  q  q        p 

A  (u*  A  u}  =  — — »     Aoit  =  — • 

v^ql  v  ql      vql 

Because  of  the  identity  (V,  38) 

we  have  also 

(16)  A2^  =  -^-- 

If  the  last  of  equations  (13)  be  divided  by  qlt  and  the  values  of 
'i»  Pi/2i  obtained  from  (15)  and  (16)  be  substituted,  we  have 


22        t»3    2        i'4          2  v 
In  consequence  of  the  first  of  equations  (15),  written 

(18)  v  =  -L=, 

VAjt* 

the  coefficients  of  f,  TJ,  fv,  rjl  in  (17)  are  expressible  in  terms  of 
differential  parameters  of  u  formed  with  respect  to  (14),  as  was 
to  be  proved. 

An  exceptional  case  is  that  in  which  q^  0.  Under  this  condition 
the  spherical  representation  of  the  z-axis  reduces  to  a  curve,  as  is 
seen  from  (14). 

*  Previously  we  have  indicated  by  a  prime  differential  parameters  formed  with  respect 
to  the  linear  element  of  the  spherical  representation.  For  the  sake  of  simplicity  we  dis 
regard  this  practice  in  this  chapter. 


THEOREM  OF  WE1NGAKTEN  355 

By  means  of  (9)  we  find  that 

(19)  A,  (A,,  tO  =  ^.       M^,  «)  =  ^f.       \(Z»  «)  =  f' 
and  consequently  equations  (11)  may  be  written 

(20)  x  =[^  +  W\  (X»  *>)]  du  +  [f  ^  +  v,v\  (X»  u)]  dv, 


and  similarly  for  y  and  2. 

147.  The  theorem  of  Weingarten.  Equation  (17)  is  the  equation 
which  Weingarten  has  suggested  as  a  substitute  for  equation  (1). 
We  notice  that  f,  ?;,  fx,  T/I  are  known  functions  of  u  and  v  when 
the  surface  S  is  given.  By  means  of  (18)  equation  (17)  can  be 
given  a  form  which  involves  only  u  and  differential  parameters 
of  u  formed  with  respect  to  (14).  On  account  of  the  invariant 
character  of  these  differential  parameters  this  linear  element 
may  be  expressed  in  terms  of  any  parameters,  say  u'  and  v'. 
We  shall  show  that  each  solution  of  equation  (17)  determines 
a  surface  applicable  to  S.  We  formulate  the  theorem  of  Wein 
garten  as  follows  : 

Let  S  be  a  surface  whose  linear  element  in  the  reduced  form  is 
(21  )        ds*  =  (?2  +  T?2)  du*  +  2  (^  +  wj  dudv  +  (tf  + 
then  %,  rj,  fj,  ?;1  are  functions  of  u  and  v  such  that 


Furthermore,  let  Xv  Yv  Z^  le  the  coordinates  of  a  point  on  the  unit 
sphere,  expressed  in  terms  of  any  two  parameters  u1  and  v',  the  linear 
element  of  the  sphere  being 

(23)  da1'2  =  &  du'*  +  2  &'  du'dv'  +  £>dv'\ 
Any  integral  ul  of  the  equation 

(24)  Ju,  —L   AMu  -  Ju,  -l= 


-      u, 


A,w 


t«)=  0, 


356  DEFORMATION  OF  SURFACES 

the  differential  parameters  being  formed  with  respect  to  (23),  renders 
the  following  expression  and  similar  ones  in  y  and  z  total  differentials: 

(25) 


where 


f  Ae  surface  whose  coordinates  are  the  functions  x,  y,  z  thus  defined 
has  the  linear  element  (21). 

Before  proving  this  theorem  we  remark  that  the  parameters  u' 
and  v'  may  be  chosen  either  as  known  functions  of  u  and  v,  or  in 
such  a  way  that  the  linear  element  (14)  shall  have  a  particular 
form.  In  the  former  case  X^  Yv  Z^  are  known  as  functions  of  uf 
and  v',  and  in  the  second  their  determination  requires  the  solution 
of  a  Riccati  equation.  However,  in  what  follows  we  assume  that 
Xv  Yv  Zl  are  known. 

Suppose  now  that  ur  and  vf  are  any  parameters  whatever,  and 
that  we  have  a  solution  u^  of  equation  (24),  where  the  differential 
parameters  are  formed  with  respect  to  (23).  Let  v^  denote  the 
quantity  (A^)"*.  Both  u^  and  vl  are  functions  of  u'  and  v',  and 
consequently  the  latter  are  expressible  as  functions  of  the  former. 
We  express  X^  Y^  Z{  as  functions  of  ul  and  vl  and  determine  the 
corresponding  linear  element  of  the  unit  sphere,  which  we  write 

(26)  dffl  =  (£;  dul  +  2  ^  dujvt  +  ^  c(v*. 

In  terms  of  ul  and  vl  we  have 


,  i 

From  these  expressions  it  follows  that  if  we  put 


we  have 
(27) 


METHOD  OF  WEINGARTEN  357 

Hence  if  we  put 

x  =  Y&  -  Z,Y»      Y  =  z&  -  x&,     z  =x1Ya  -  r^, 

the  functions  A^,  Yv  •  -  . ,  Z  satisfy  a  set  of  equations  similar  to  equa 
tions  (V,  47). 

In  consequence  of  (27)  the  corresponding  rotations  have  the  values 

dX  dX 


.dx 

r1=0. 

It  is  readily  shown  that  these  functions  satisfy  equations  similar 
to  (10). 

Since  the  functions  f,  77,  £,  ^  are  of  the  same  form  in  (25)  as 
in  (21),  equations  similar  to  the  first  two  of  equations  (13)  are  neces 
sarily  satisfied.  Hence  the  only  other  equation  to  be  satisfied,  in 
order  that  the  expressions  (25)  be  exact  differentials,  is 

But  it  can  be  shown  that  the  coefficients  of  (26)  are  expressible  in 
the  form  g  _  v*\  ^  <%•  __  —  ^  _  -2 

so  that  by  means  of  differential  parameters  of  u^  formed  with 
respect  to  (26)  the  equation  (28)  can  be  given  the  form  (17). 
Hence  all  the  conditions  are  satisfied,  and  the  theorem  of  Wein- 
garten  has  been  established. 

148.  Other  forms  of  the  theorem  of  Weingarten.  It  is  readily 
found  that  equations  (22)  are  satisfied  by  the  expressions 


(29) 


dv 


du  dv  dv 

where  (/>  is  any  function  of  u  and  v.    Since  now 
(30)  ,7  +  ^=0, 

equation  (17)  reduces  to 


358  DEFOKMATION  OF  SURFACES 

This  equation  will  be  simplified  still  more  by  the  introduction 
of  two  new  parameters  which  are  suggested  by  the  following 
considerations. 

As  previously  defined,  the  functions  X^  Yv  Z^  are  the  direction- 
cosines  of  lines  tangent  to  the  given  surface  S  in  such  a  way  that 
the  ruled  surface  formed  by  these  tangents  at  points  of  a  curve 
u  =  const,  has  this  curve  for  its  line  of  striction.  Moreover,  from 
the  theorem  of  Weingarten  it  follows  that  the  functions  X^  Yv  Zl 
have  the  same  significance  for  the  surface  applicable  to  S  which 
corresponds  to  a  particular  solution  of  equation  (17). 

But  Xv  Yv  Zl  may  be  taken  also  as  the  direction-cosines  of  the 
normals  to  a  large  group  of  surfaces,  as  shown  in  §  67.  In  partic 
ular,  we  consider  the  surface  S  which  is  the  envelope  of  the  plane 

$  +  Zj,  =  u. 


Each  solution  of  equation  (17)  determines  such  a  surface.  If  x,  y,  z 
denote  the  coordinates  of  the  point  of  contact  of  this  plane  with  S, 
we  have  from  (V,  32) 

(32)  x  =  uXl+\(u,X1), 
which,  in  consequence  of  (19),  may  be  written 

(32')  i  =  wX-f-X. 

v 

Hence  the  point  of  contact  of  S  lies  in  the  plane  through  the  origin 
parallel  to  the  tangent  plane  to  S  at  the  corresponding  point. 

If  the  square  of  the  distance  of  the  point  of  contact  from  the 
origin  be  denoted  by  2  ^,  and  the  distance  from  the  origin  to  the 
tangent  plane  by  p,*  we  have 

(33)  2g  =  sa+ya+ia=wa+^.         p  =  u. 

From  (V,  35,  37)  it  follows  that  the  principal  radii  of  2  are 
given  by 
(34) 


*  The  reader  will  observe  that  the  functions  p  and  q  thus  defirfed  are  different  from 
the  rotations  designated  by  the  same  letters.  As  this  notation  is  generally  employed  in 
the  treatment  of  the  theorem  of  Weingarten,  it  has  seemed  best  to  retain  it,  even  at  the 
risk  of  a  confusion  of  notation. 


METHOD  OF  WEINGARTEN 


359 


where  the  differential  parameters  are  formed  with  respect  to  (14). 
From  these  equations  we  have 


(35) 


We  shall  now  effect  a  change  of  parameters,  using  p  and  q 
defined  by  (33)  as  the  new  ones.    By  direct  calculation  we  obtain 


y^r  *^r    I    ^*r  „.  ^*r v*r 

o - -  ^  - _  o  _   -i   ^  ^ . -  -.;?    o 


(36) 


du      dp      dq  dv          v*  dq 


dif      dp* 


—  +  P2  — T  +  — ~ 

a/?a^        ^2     a^ 

_i — z.  /^ 

'      Q^,2-r 


cudv          vdpcq 

^-1^      1 

dv2  ~  v6  dq2      tf 


By  means  of  the  equations  (33)  and  (36)  the  fundamental  equa 
tion  (31)  can  be  reduced  to 


(37) 


This  is  the  form  in  whicli  the  fundamental  equation  was  first  con 
sidered  by  Weingarten.*  The  method  of  §§  146,  147  was  a  subse 
quent  development. 

In  terms  of  the  parameters  p  and  q  the  formulas  (29)  become 


dpdq          dq 


(38) 


If  these  values  and  the  expression  for  \(u,  X^)  given  by  (32)  be 
substituted  in  (20),  it  is  reducible  to 


dp*         dpdq/  "       \    "tip'tiq         cq' 
*  Comptes  Rendus,  Vol.  CXII  (1891),  p.  607. 


360  DEFORMATION  OF  SURFACES 

Hence  the  equations  for  S  may  be  written 


(39) 


and  consequently  the  linear  element  of  S  is  of  the  form 


Since  these  various  expressions  and  equations  differ  only  in  form 
from  those  which  figure  in  the  theorem  of  Weingarten,  the  latter  is 
just  as  true  for  these  new  equations.  We  remark  also  that  the  right- 
hand  member  of  (40)  depends  only  upon  the  form  of  c£.  Hence  we 
have  the  theorem  of  Weingarten  in  the  form  : 

When  (j)  in  equation  (37)  is  a  definite  function  of  p  and  q,  this 
equation  defines  a  large  group  of  surfaces  with  the  same  spherical 
representation,  the  functions  pl  and  p2  denoting  the  principal  radii, 
and  p  and  2q  the  distance  from  the  origin  to  the  tangent  plane  and 
the  square  of  the  distance  to  the  point  of  contact.  Each  surface  2 
satisfying  this  condition  gives  by  quadratures  (39)  a  surface  with  the 
linear  element  (40).  Conversely,  each  surface  with  this  linear  element 
stands  in  such  relation  to  some  surface  satisfying  the  corresponding 
equation  (37). 

As  a  corollary  to  the  preceding  results,  we  have  the  theorem  : 

The  linear  element  of  any  surface  S  is  reducible  to  the  form 

(41)  di~  =  du*  +  2  ^  dudv  +  2  ^  dv\ 

du  dv 

where  ^r  is  a  function  of  u  and  v. 

For,  we  have  seen  that  the  linear  element  of  any  surface  is 
reducible  to  the  form  (40).  If,  then,  we  change  the  parameters  by 
means  of  the  equations 


we  have 

(43)  ds*  =  du*+2p  dudv  +  2q  dv\ 


METHOD  OF  WEINGARTEN                          361 
From  (42)  it  follows  that 

a20  ,         32<t>    ,  tf$    ,     ,  d24>  , 

<fu  =  —  ^  dp  +  —  -i-  <fy,  dt>  =  —i-  dp  +  —  ^  rfg, 

dp1           ejpdg  dp  £3           a/ 

and  consequently 

A 


(44) 
where 


A  = 


dp2  dq2      \dp  dq 


From  (44)  it  is  seen  that  ~-  =-^j  and  consequently  the  inverse 

dv      cu 

of  equations  (42)  are  of  the  form 

fAfii  W  W 

(45)  »  =  -X,          q  =  ^L. 

du  *       dv 

Hence  equation  (43)  is  of  the  form  (41),  as  was  to  be  proved. 
Moreover,  equations  (44)  reduce  to 


(46)  =       A          -=   ----  A 

2  dpdq 


In  terms  of  these  parameters  w,  v  equations  (39)  reduce  to 

(47)  dx  =  X^du  +  ^c?v,     (#?/  =  I^c^tt  +  ^^v,     dz  =  Z^du  -f  ^c?v. 

Hence  the  coordinates  of  2  are  given  by 

AQ  _       ^C  _       3v  ^2 

(48)  X  =  T^         y  =  -£-J         z  =  ^"f 

dv  dv  dv 

and  the  direction-cosines  of  the  normal  to  2  are 

(49)  X.A       rl  =  ^,       Z1==^, 

aw  aw  ^ 

that  is,  the  normals  to  2  are  parallel  to  the  corresponding  tangents 
to  the  curves  v  =  const,  on  S.   Hence  we  have  the  following  theorem  : 

When  the  linear  element  of  a  surface  is  in  the  form  (41),  the  sur 
face  2  whose  coordinates  are  given  by  (48)  has  the  same  spherical 


362  DEFORMATION  OF  SURFACES 

representation  of  its  normals  as  the  tangents  to  the  curves  v  =  const. 
on  S.  If  p  and  2q  denote  the  distance  from  the  origin  to  the  tangent 
plane  to  S  and  the  square  of  the  distance  to  the  point  of  contact,  they 
have  the  values  (45).  Moreover,  if  the  change  of  parameters  defined 
by  these  equations  be  expressed  in  the  inverse  form 

/cn,  d<f>  d(f> 

(50)  M  =  s   '        v=*   ' 

dp  dq 

the  principal  radii  of  2  satisfy  the  condition 


and  the  coordinates  of  S  are  given  by  quadratures  of  the  form 
(52)  dx 


Moreover,  ever//  surface  with  the  same  representation  as  2,  and  whose 
functions  pv  /?2,  p,  q  satisfy  (51)  for  the  same  (f>,  determines  by  equa 
tions  of  the  form  (52)  a  surface  applicable  to  S.* 

149.  Surfaces  applicable  to  a  surface  of  revolution.  When  the 
linear  element  of  a  surface  applicable  to  a  surface  of  revolution 
is  written 

(53)  d?=du*  +  p*(ul)dv*, 

and  the  z-axis  of  the  moving  trihedral  is  tangent  to  the  curve 
v  —  const.,  the  function  r  is  equal  to  zero,  as  follows  from  (4). 

In  order  to  obtain  the  conditions  (8),  we  effect  the  transformation 
of  variables 

u  =  v^         v  =  —  u^ 

so  that  the  linear  element  becomes 

(54)  ds*  =  p2du*+dv2. 

Now  r  =  pf,  7^=0,  and  consequently  in  order  to  have  the  linear 
element  in  the  reduced  form  we  must  take 

(55)  u  =  u,         v=p'(—v). 

*  For  a  direct  proof  of  this  theorem  the  reader  is  referred  to  a  memoir  by  Goursat, 
Sur  un  theoreme  de  M.  Weingarten,  et  sur  la  the'orie  des  surfaces  applicables,  Toulouse 
Annales,  Vol.  V  (1891)  ;  also  Darboux,  Vol.  IV,  p.  316,  and  Bianchi,  Vol.  II,  p.  198. 


SURFACES  OF  REVOLUTION          363 

From  these  results  and  (32')  we  find  that  the  coordinates  of  the 
surface  2  are  given  by 


pf  du^      p  dvl  p'  cuv      p  dvl 

i==_lJl  +  !!l.^, 

p'  cu^      p  dVl 

and  the  direction-cosines  of  the  normals  to  2  are 

Y     1  fo          v     1  fy          ~      1  8* 

.A.,  =  --  •>  JL  1  —  --  5  Zs*  =  --  • 

p  dvl  p  cvl  p  dv1 

Also,  we  have 

(56)  P=^xX^vv         2?=^2=^  +  ~ 

Hence  we  have  the  theorem  : 

To  a  curve  which  i*  the  deform  of  a  meridian  of  a  surface  of  revo 
lution  there  corresponds  on  the  surface  2  a  curve  such  that  the  tangent 
planes  to  2  at  points  of  the  curve  are  at  a  constant  distance  from  the 
origin,  and  to  a  deform  of  a  parallel  there  corresponds  a  curve  such 
that  the  projection  of  the  radius  vector  upon  the  tangent  plane  at  a 
point  is  constant. 

For  the  present  case  77  =  f  1  =  0  ;  consequently  we  have,  from  (38), 


Sf 

This  equation  is  satisfied  by 

(57)  </>(£>,  #)=/(2  q—  jt?2), 

where  /  is  any  function  whatever.    In  terms  of  this  function  we 

have,  from  (38), 

where    the    accents    indicate   differentiation  with  respect  to  the 
argument,   2q—p2. 

By  means  of  (55)  the  linear  element  (54)  can  be  transformed  into 


the  function  a)(v)  being  defined  by 


364 


DEFORMATION  OF  SURFACES 


Since  77  =  f  l  =  0,  we  have 


and  we  know  that  r  =  v.    Now  equations  (58)  become 

and  these  are  consistent  because  of  the  relation  2^  —  p~  =  \/v2, 
which  results  from  (56).    Hence  we  have  the  theorem : 

When  <£  (p,  q)  is  a  function  of  2  q  —  p\  the  corresponding  surface  S 
is  applicable  to  a  surface  of  revolution,  the  tangents  to  the  deforms  of 
the  parallels  being  parallel  to  the  corresponding  normals  to  2. 

If  we  give  <j>  the  form  (57)  and  put  ^  =  2f,  the  linear  element 
of  Sis 

(59)  ds2  =  (^q  — p2)  d^2  +  ^fr2dp2, 

as  follows  from  (40)  or  (58). 

150.  Minimal  lines  on  the  sphere  parametric.  In  §  147  we  re 
marked  that  the  parametric  curves  on  the  sphere  may  be  any  what 
ever.  An  interesting  case  is  that  in  which  they  are  the  imaginary 
generatrices.  In  §  35  we  saw  that  the  parameters  of  these  lines, 
say  a  and  /3,  can  be  so  chosen  that 

a/3-1 


(60)  X,= 

Consequently 
(61) 


-  +  a/3 
da2=dX?  + 


a/3 


rp    


4  dad /3 


From  (32)  we  find  that  the  coordinates  of  2,  the  envelope  of 
the  plane  Xx  +  Yy+Zz  —  p  =  0 

are 


(62) 


z  = 


From  these  we  obtain 
(63)  2 


MINIMAL  LINES  ON  THE  SPHEKE 


365 


By  means  of  (34)  the  expressions  for  pl+  p2  and  p^  in  terms 
of  p  and  its  derivatives  with  respect  to  a  and  yS  can  be  readily 
found,  and  thus  the  fundamental  equation  (37)  put  in  a  new  form. 
However,  it  is  not  with  the  general  case  that  we  shall  now  concern 
ourselves,  but  with  a  particular  form  of  the  function  <j>(p,  q). 

This  function  has  been  considered  by  Weingarten  * ;  it  is 


(64) 

In  this  case 


so  that  equation  (37)' reduces  to 

(65)  />i  +  ft  =  -(2;>  + 

which,  in  consequence  of  (34),  may  be  written 


(66) 


'(P) 


dad  j3      (l  +  a/3)2 
When  the  values  from  (62)  are  substituted  in  (52),  we  obtain 


(67) 

where 
(68) 


z  =  *&  -  Cu1dZl  +   « 
J  \ 


occ 


From  (42)  and  (64)  we  have 

u  —  q—p2—a>f  ( p),         v=p. 

Hence  the  linear  element  (43)  of  *S  is,  in  this  case, 
(69)  ds2  =du2+2v  dudv  +  2  [u  +  v2  +  w'(v)]  dv2. 

*Acta  Mathematica,  Vol.  XX  (1896),  p.  195. 


366  DEFORMATION  OF  SURFACES 

However,  from  (68)  it  is  seen  that 

v2 

(70)  Ul=u  +  -, 

so  that  (69)  may  be  written 

(71)  ds2  =  dul  +  2  [MI  +  ®'(t>)]  *>a. 
Gathering  together  these  results,  we  have  the  theorem  : 

2%e  determination  of  all  the  surfaces  ivith  the  linear  element  (71) 
reduces  to  the  integration  of  the  equation 


The  integral  of  this  equation  for  o»(_p)  arbitrary  is  not  known. 
However,  the  integral  is  known  in  certain  cases.  We  consider 
several  of  these. 

151.  Surfaces  of  Goursat.  Surfaces  applicable  to  certain  parab 
oloids.  When  we  take 

(73)  v'(p)=im(l-m)p\ 

m  being  any  constant,  equation  (72)  becomes 

m(l-m)p 


dad  ft        (l  +  aj3f 

The  general  integral  of  this  equation  can  be  found  by  the  method 
of  Laplace,*  in  finite  form  or  in  terms  of  definite  integrals,  accord 
ing  as  m  is  integral  or  not. 

The  linear  element  of  the  surface  S  is 

(75)  ds1  =  du?  +  [2  u^  H-  m  (1  -  m)  ^]  dv\ 
And  the  surfaces  2  are  such  that 

(76)  p1  +  pt+2p  =  m(m-I)p, 

that  is,  the  sum  of  the  principal  radii  is  proportional  to  the  dis 
tance  of  the  tangent  plane  from  a  fixed  point.  These  surfaces 
were  first  studied  by  Goursat,  f  and  are  called,  consequently,  the 
surfaces  of  G-  our  sat. 

*Darboux,  Vol.  II,  p.  66.  t  American  Journal,  Vol.  X  (1888),  p.  187. 


SURFACES  OF  GOURSAT  367 

Darboux  has  remarked*  that  equation  (71)  is  similar  to  the  linear 
element  of  ruled  surfaces  (VII,  53).  In  fact,  if  the  equations  of  a 
ruled  surface  are  written  in  the  form 


(77)  x  =  x0+lul,         y 

where  #0,  •  •  •  ;  /,  w,  n  are  functions  of  v  alone,  which  now  is  not 
necessarily  the  arc  of  the  directrix,  the  linear  element  of  the 
surface  will  have  the  form  (71),  provided  that 

(78)  2J2  =  1,    2a^  =  0,    2X2  =  2  w'(^),    2a#'  =  l,    2/'2=0. 

In  consequence  of  the  equations 
2ft'  =0, 


it  follows  that  a  ruled  surface  of  this  kind  admits  an  isotropic 
plane  director.    If  this  plane  be  x  +  iy  =  0,  that  is,  if 

we  have 

where  V  is  a  function  of  v.    By  means  of  these  values  and  equa 
tions  (78),  we  can  put  (77)  in  the  form 

*dv 


(79) 


=  %Vu^+%  Cv'v'dv  -  C~  dv, 

/y 
ytdv- 


We  shall  find  that  among  these  surfaces  there  is  an  imaginary 
paraboloid  to  which  are  applicable  certain  surfaces  to  which  Wein- 
garten  called  attention.  To  this  end  we  consider  the  function 

_2p^ 

(80)  ®'(_p)=:—  ^ficp  —  2  tee  v" 

where  K  denotes  a  constant.    Now  equation  (66)  becomes 

_2£ 

A          Vic  _  1 

—' 


*  Vol.  IV,  p.  333. 


368  DEFOKMATION  OF  SURFACES 

In  consequence  of  the  identity 


the  preceding  equation  is  equivalent  to 
log(l  +  a/3)V^  =  - 


dad/3 

€ 

If  we  put  JL 

this  equation  takes  the  Liouville  form 


-20 
> 


'dad  IB 
of  which  the  general  integral  is 

1      A'S' 


where  A  and  B  are  functions  of  a  and  /3  respectively,  and  the 
accents  indicate  differentiation  with  respect  to  these.  Hence  the 
general  integral  of  (81)  is 

c%=      2^  +  AS) 
VAB'(l  +  a/3) 

and  the  linear  element  of  S  is 


(82)  ds2=  du^2\ul-  V^K  -  2 

If  now,  in  addition  to  (80),  we  take 

V 


the  equations  (79)  take  such  a  form  that 
(83)  (x+iy)x  =  —  icz. 

Hence  the  surfaces  with  the  linear  element  (82)  are  applicable  to 
the  imaginary  paraboloid  (83).  The  generator  x  +  iy  =  Q  of  this 
paraboloid  in  the  plane  at  infinity  is  tangent  to  the  imaginary 
circle  at  the  point  (x:y:z  =  l:i:Q),  which  is  a  different  point 
from  that  in  which  the  plane  at  infinity  touches  the  surface, 
that  is,  the  point  of  intersection  of  the  two  generators. 


DEFOBMATION  OF  PARABOLOIDS  369 

Another  interesting  case  is  afforded  when  m  in  (73)  has  the 
value  2.    Then  u>'(v)  =  —  v2,  and  equation  (71)  becomes 


(84)  ds2  =  duf  +  2  (Ml  - 

If  we  take  V=v/^/2^c,  we  obtain  from  equations  (79) 


from  which  we  find,  by  the  elimination  of  ul  and  v, 
(85)  (z  +  i»  2  =  K  (x  —  it/)  . 

The  generator  x  +  iy  =  0  in  the  plane  at  infinity  on  the  paraboloid 
(85)  is  tangent  to  the  imaginary  circle  at  the  point  (x:  y:  z  —  1  :  i:  0), 
just  as  in  the  case  of  the  paraboloid  (83),  but  the  paraboloid  (85)  is 
tangent  to  the  plane  at  infinity  at  the  same  point. 

GENERAL  EXAMPLES 

1  .  A  moving  trihedral  can  be  associated  with  a  surface  in  an  infinity  of  ways  so 
that  as  the  vertex  of  the  trihedral  describes  a  curve  u  =  const,  the  z-axis  generates 
a  ruled  surface  whose  line  of  striction  is  this  curve. 

2.  The  tangents  to  the  curves  v  =  const,  on  a  surface  at  the  points  where  these 
curves  are  met  by  an  integral  curve  of  the  equation 


form  a  ruled  surface  for  which  the  latter  curve  is  the  line  of  striction. 

3.  If  the  ruled  surface  formed  by  an  infinity  of  tangents  to  a  surface  S  has  the 
locus  of  the  points  of  contact  for  its  line  of  striction,  this  relation  is  unaltered  by 
deformations  of  S. 

4 .  Show  that  if  D,  D',  D"  are  the  second  fundamental  coefficients  of  a  sur 
face  with  the  linear  element  (53),  the  equation  of  the  lines  of  curvature  of  the 
associated  surface  S  is  reducible  to 

Ddii!  +  D'dvi    V'dui  +  D"d 


—;  dui  pp'  dv\ 

p 


5  .  Show  that  the  surface  S  associated  by  the  method  of  Weingarten  with  a  sur 
face  S  applicable  to  a  surface  of  revolution  corresponds  with  parallelism  of  tangent 
planes  to  the  surface  S'  complementary  to  S  with  respect  to  the  deforms  of  the 
meridians  ;  and  that  the  lines  of  curvature  on  S  and  S'  correspond. 


370  DEFORMATION  OF  SURFACES 

6.  Show  that  when  0  has  the  form  (57),  the  equation  (51)  is  reducible  to 


hence  the  determination  of  all  the  surfaces  applicable  to  surfaces  of  revolution  is 
equivalent  to  the  determination  of  those  surfaces  S  which  are  such  that  if  MI  and 
M2  are  the  centers  of  principal  curvature  of  2  at  a  point  I/,  and  N  is  the  projection 
of  the  origin  0  on  the  normal  at  M,  the  product  NMi  •  NM2  is  a  function  of  ON. 

7.  Given  any  surface  S  applicable  to  a  surface  of  revolution.    Draw  through  a 
fixed  point  O  segments  parallel  to  the  tangents  to  the  deforms  of  the  meridians 
and  of  lengths  proportional  to  the  radii  of  the  corresponding  parallels,  and  through 
the  extremities  of  these  segments  draw  lines  parallel  to  the  normals  to  S.    Show 
that  these  lines  form  a  normal  congruence  whose  orthogonal  surfaces  2  have  the 
same  spherical  representation  of  their  lines  of  curvature  as  S  and  are  integral  sur 
faces  of  the  equation  of  Ex.  6. 

8.  Let  fi  be  a  surface  applicable  to  a  surface  of  revolution  and  S'  the  surface 
complementary  to  S  with  respect  to  the  deforms  of  the  meridians  ;  let  also  S  and 
S'  be  surfaces  associated  with  S  and  S'  respectively  after  the  manner  of  Ex.  7. 
Show  that  corresponding  normals  to  S  and  S'  are  perpendicular  to  one  another, 
and  that  the  common  perpendicular  to  these  normals  passes  through  the  origin  and 
is  divided  by  it  into  two  segments  which  are  functions  of  one  another. 

9.  Show  that  a  surface  determined  by  the  equation 

2  q  +  K  +  (PI  +  pz)p  +  PIPZ  =  0, 

where  K  is  a  constant,  possesses  the  property  that  the  sphere  described  on  the  seg 
ment  of  each  normal  between  the  centers  of  principal  curvature  with  this  segment 
for  diameter  cuts  the  sphere  with  center  at  the  origin  and  of  radius  V±  K  in  great 
circles,  orthogonally,  or  passes  through  the  origin,  according  as  K  is  positive,  nega 
tive,  or  zero.  These  surfaces  are  called  the  surfaces  of  Bianchi. 

10.  Show  that  for  the  surfaces  of  Bianchi  the  function  0(p,  q)  is  of  the  form 


.  0  =  V2  q  -  p2  +  /c, 

and  that  the  linear  element  of  the  associated  surface  S  applicable  to  a  surface 
of  revolution  is  1 


Show  also  that  according  as  /c  =  0,  >  0,  or  <  0  the  linear  element  of  S  is  reducible 
to  the  respective  forms 

ds2  =  dw2  +  e2Mdv2,  ds2  =  tanh4  u  du2  +  sech2w  du2,  ds2  =  coth4  u  du2  -f  csch2u  dv2. 
On  account  of  this  result  and  Ex.  10,  p.  318,  the  surfaces  of  Bianchi  are  said  to  be 
of  the  parabolic,  elliptic,  or  hyperbolic  type,  according  as  K  =  0,  >  0,  or  <  0. 

11.  Let  S  be  a  pseudospherical  surface  with  its  linear  element  in  the  form 
(VIII,  32),  and  Si  the  Bianchi  transform  whose  linear  element  is  (VIII,  33).  Find 
the  coordinates  x,  y,  z  of  the  surface  S  associated  with  Si  by  the  method  of  Wein- 
garten,  and  show  that  by  means  of  Ex.  8,  p.  291,  the  expression  for  x  is  reducible  to 

| 
x  =  aea  (cos  6X1  +  sin  0JT2)  4-  fX, 

where  X\,  JT2,  X  are  the  direction-cosines  with  respect  to  the  sc-axis  of  the  tangents 
to  the  lines  of  curvature  of  S  and  of  the  normal  to  the  latter. 


GENERAL  EXAMPLES  371 

12.  Show  that  the  surfaces  S  and  S  of  Ex.  11  have  the  same  spherical  represen 
tation  of  their  lines  of  curvature,  that  S  is  a  surface  of  Bianchi  of  the  parabolic 
type,  and  that  consequently  there  is  an  infinity  of  these  surfaces  of  the  parabolic 
type  which  have  the  same  spherical  representation  of  their  lines  of  curvature  as  a 
given  pseudospherical  surface  S. 

13.  Show  that  if  Si  and  S2  are  two  surfaces  of  Bianchi  of  the  parabolic  type 
which  have  the  same  spherical  representation  of  their  lines  of  curvature,  the  locus 
of  a  point  which  divides  in  constant  ratio  the  line  joining  corresponding  points  of 
Si  and  S2  is  a  surface  of  Bianchi  with  the  same  representation  of  its  lines  of  cur 
vature,  and  that  it  is  of  the  elliptic  or  hyperbolic  type  according  as  the  point  divides 
the  segment  internally  or  externally. 

14.  When  S  is  a  pseudospherical  surface  with  its  linear  element  in  the  form 
(VIII,  32),  the  coordinates  x~i,  yi,  z\  of  the  surface  S  determined  by  the  method 
of  Weingarten  are  reducible  to 

A  J- 

Xi  =  (ae   a  cos  6  +  y  sin  6)  X\  -f  (ae   a  sin  0  —  77  cos  0}  JF2  , 

and  analogous  expressions  for  yi  and  z1?  where  JTi,  FI,  Zx;  JT2,  F2,  Z2  are  the 
direction-cosines  of  the  tangents  to  the  lines  of  curvature  of  S.  Show  also  that  S 
has  the  same  spherical  representation  of  its  lines  of  curvature  as  the  surface  Si  with 
the  linear  element  (VIII,  33). 

15.  Derive  from  the  equations 

xXi  +  yYi  +  zZt  =p,  X2  +  £2  +  £2  =  2g, 

by  means  of  (44),  (48),  and  (49),  the  equations 


where  x,  y,  2  are  the  coordinates  of  S. 

16.  Show  that  the  equations  for  S  similar  to  (IV,  27)  are  reducible  to 


dudv  cv*  \cu*  cucv 

and  similar  expressions  in  y  and  z.    Derive  therefrom  (cf.  Ex.  15)  the  equations 
D'du  +  D"dv  +  r(Ddu  +  lYdv)  -  0, 


du  — 


dpdq  dp*  \dq2  cpdq 

where  D,  D',  D"  are  the  second  fundamental  coefficients  of  8. 

17.  Show  that  the  lines  of  curvature  on  S  correspond  to  a  conjugate  system  on 
S  (cf.  Ex.  16). 

1 8.  Show  that  for  the  surface  S  we  have 

dx  dX\  dx       dXi  .  dXi 

2p  ~      plp<*    eq  dq~    dp  '     dq 

19.  Let  S  be  the  surface  defined  by  (67)  and  Si  the  surface  whose  coordinates  are 

Xi  =  x  —  u\X\,         yi  =  y  —  WiFi,         z\  —  z  —  u\L\. 

Show  that  Si  is  an  involute  of  /S,  that  the  curves  p  =  const,  are  geodesies  on  S  and 
lines  of  curvature  on  Si,  and  that  the  radii  of  principal  curvature  of  Si  are 


372  DEFORMATION  OF  SURFACES 

20.  Show  that  when  m  in  (73)  is  0  or  1,  the  function  p  is  the  sum  of  two  arbi 
trary  functions  of  a  and  /3  respectively,  that  the  linear  element  of  S  is 

ds*=  dw12  +  2i*idw2, 

that  S  is  an  evolute  of  a  minimal  surface  (cf.  Ex.  19),  and  that  the  mean  evolute  of 
S  is  a  point. 

21.  Show  that  when  m  in  (73)  is  2,  the  general  integral  of  equation  (74)  is 


where  /i  and  /2  are  arbitrary  functions  of  a  and  0  respectively.    Show  also  that  the 
surface  2  is  minimal  (cf.  §  151). 

22.  Show  that  the  mean  evolute  of  a  surface  of  Goursat  is  a  surface  of  Goursat 
homothetic  to  the  given  one. 

23.  Show  that  when  u>(p)  =  ^op2,  then 

p  =  a  log(l  +  a/3)  +/i(nr)  +/2(/3), 
where  /i  and  /2  are  arbitrary  functions,  that  the  linear  element  of  S  is 

ds2  =  du?  +  2  (ui  +  aw)  du2, 
and  that  the  mean  evolute  of  2  is  a  sphere. 

24.  Show  that  the  surfaces  S  of  Ex.  23  are  applicable  to  the  surfaces  of  revolu 
tion  S0  whose  equations  are 

v  .    v 

0  a '  a ' 

where  a  is  an  arbitrary  constant.    Show  also  that  when  a  =  ia,  £0  is  a  paraboloid. 

25.  Show  that  when 


c   /~17 
=  I   vu*  —  a'2  — 

J 


the  surfaces  S  are  spherical  or  pseudospherical  according  as  m  is  positive  or  nega 
tive  ;  also  that  the  surfaces  <S  are  applicable  to  the  surface 

1)2  M2 

x  +  iy  =  v,         x-iy  =  —  —  TT~~  mu'        ''z  =  w» 
2       2m 

which  is  a  paraboloid  tangent  to  the  plane  at  infinity  at  a  point  of  the  circle  at 
infinity. 


CHAPTER  XI 

INFINITESIMAL  DEFORMATION  OF  SURFACES 

152.  General  problem.  The  preceding  chapters  deal  with  pairs 
of  isometric  surfaces  which  are  such  that  in  order  that  one  may 
be  applied  to  the  other  a  finite  deformation  is  necessary.  In 
the  present  chapter  we  shall  be  concerned  with  the  infinitesimal 
deformations  which  constitute  the  intermediate  steps  in  such  a 
finite  deformation. 

Let  x^y,z\  x',  y\  z'  respectively  be  the  coordinates  of  a  surface 
S  and  a  surface  S\  the  latter  being  obtained  from  the  former  by  a 
very  small  deformation.  If  we  put 

(1)  x'^x  +  ex^,          yf=y  +  cyv          z'^z  +  ez^ 

where  e  denotes  a  small  constant  and  xv  y^  zl  are  determined  func 
tions  of  u  and  v,  these  functions  are  proportional  to  the  direction- 
cosines  of  the  line  through  corresponding  points  of  S  and  Sf.  From 
these  equations  we  have 

dx'*  +  dy'*  +  dzn  =  da?  +  dy2  +  dz2  +  2e(dx  dxl  +  dy  dyl  +  dz  dzj 


If  the  functions  satisfy  the  condition 

(2)  dx  dxl  -f  dy  dy^  +  dz  dzl  =  0, 

corresponding  small  lengths  on  S  and  Sr  are  equal  to  within  terms 
of  the  second  order  in  e.  When  e  is  taken  so  small  that  e2  may  be 
neglected,  the  surface  S'  defined  by  (1)  is  said  to  arise  from  S  by 
an  infinitesimal  deformation  of  the  latter.  In  such  a  deformation 
each  point  of  S  undergoes  a  displacement  along  the  line  through 
it  whose  direction-cosines  are  proportional  to  xv  yv  zr  These  lines 
are  called  the  generatrices  of  the  deformation. 

It  is  evident  that  the  problem  of  infinitesimal  deformation  is 
equivalent  to  the  solution  of  equation  (2).     Since  xv  y^  zl  are 

373 


374  INFINITESIMAL  DEFORMATION 

functions  of  u  and  v,  they  may  be  taken  for  the  coordinates  of  a 
surface  £r  Equation  (2)  expresses  the  fact  that  the  tangent  to 
any  curve  on  S  is  perpendicular  to  the  tangent  to  the  correspond 
ing  curve  on  Sl  at  the  homologous  point.  We  say  that  in  this  case 
S  and  Sl  correspond  with  orthogonality  of  corresponding  linear  ele 
ments.  And  so  we  have: 

The  problem  of  the  infinitesimal  deformation  of  a  surface  S  is 
equivalent  to  the  determination  of  the  surfaces  corresponding  to  it 
with  orthogonality  of  linear  elements. 

153.  Characteristic  function.  We  proceed  to  the  determination 
of  these  surfaces  Sv  and  to  this  end  replace  equation  (2)  by  the 
equivalent  system 


Weingarten  *  replaced  the  last  of  these  equations  by  the  two 
^  ex  fix.  ^-v  dx  dxl 

(4)  Sssw-**    X^w=~*7/' 

thus  denning  a  function  $,  which  Bianchi  has  called  the  character 
istic  function  ;  as  usual  H  =  V  'EG  —  F*. 

If  the  first  of  equations  (3)  be  differentiated  with  respect  to  v, 
and  the  second  with  respect  to  M,  we  have 


-- 

dudv      **  fa  fru  $v  fa  du  dv        *  dv  du  dv 


With  the  aid  of  these  identities,  of  the  formulas  (V,  3),  and  of  the 
Gauss  equations  (V,  7),  the  equations  obtained  by  the  differentia 
tion  of  equations  (4)  with  respect  to  u  and  v  respectively  are 
reducible  to  . 


H 


v  H 

*  Crelle,  Vol.  C  (1887),  pp.  296-310. 


CHARACTERISTIC  FUNCTION 


375 


Excluding  the  case  where  S  is  a  developable  surface,  we  solve  these 

<r-\      dx.     TT-\      dx. 
equations  for  >,^— '  2*X-zr  anc*  obtain 

*-l     du    ^      dv 


(5) 


cu 


u 


dv 


KH 


dv 


where  K  denotes  the  total  curvature  of  S.  If  we  solve  equations 
(3),  (4),  and  (5)  for  the  derivatives  of  x^  y^  zl  with  respect  to  u 
and  v,  we  obtain 


(6) 


v 


cu 


CU 


dv 


KH 


dv 


dv 


KH 


and  similar  expressions  in  y^  and  zr  Hence,  when  the  characteristic 
function  is  known,  the  surface  Sl  can  be  obtained  by  quadratures. 
Our  problem  reduces  therefore  to  the  determination  of  <£. 

If  equations  (5)  be  differentiated  with  respect  to  v  and  u  respec 
tively,  and  the  resulting  equations  be  subtracted  from  one  another, 
we  have 


—  V     1  —  —  V  — i  — 

+4  to  du       2j  fa.   fa. 


When  the  derivatives  of  Jf,  F,  Z  in  the  right-hand  member  are 
replaced  by  the  expressions  (V,  8),  the  above  equation  reduces  to 


dv 


u         d 
KH  du 


du 


dv 


KH 


H 


Bianchi  calls  this  the  characteristic  equation. 

In  consequence  of  (IV,  73,  74)  equation  (7)  is  reducible  to 


ft 


376 


INFINITESIMAL  DEFORMATION 


where  <£",  c^,  $  are  the  coefficients  of  the  linear  element  of  the 
spherical  representation  of  /S,  namely 

(9)  da*  =  &du*  +  2  ^  dudv 

and  /K 


By  means  of  (V,  27)  equation  (8)  is  reducible  to 


where  the  Christoffel  symbols  are  formed  with  respect  to  (9). 

Since  X,  Y,  Z  are  solutions  of  equations  (V,  22),  they  are  solu 
tions  of  (10),  and  consequently  also  of  equation  (7).  Therefore  the 
latter  equation  may  be  written 


dv 


dX 


du 


KH 


du 


v 


_ 


But  this  is  the  condition  of  integrability  of  equations  (6).    Hence 
we  have  the  theorem : 

Each  solution  of  the  characteristic  equation  determines  a  surface 
S^  and  consequently  an  infinitesimal  deformation  of  S. 

154.  Asymptotic  lines  parametric.  When  the  asymptotic  lines 
on  8  are  parametric,  equation  (10)  is  reducible,  in  consequence  of 
(VI,  15),  to 


where 
If  we  put 


dudv      2      dv      du      2      du      dv 


<f,  V-  ep  =  6, 


ASYMPTOTIC  LINES  PARAMETIC 


377 


e  being  +1  or  —  1  according  as  the  curvature  of  S  is  positive  or 
negative,  equation  (11)  becomes 

(12)  ** 


Since  X,  Y,  Z  are  solutions  of  (11),  the  functions 

vl  =  X  V—  e/j,          v2  =  F  V—  ep, 
are  solutions  of  (12). 


Now  equations  (6)  may  be  put  in  the  form 

e 


(13) 


du      du 


dx, 
dv 


e 

cB 

cv 


The  reader  should  compare  these  equations  with  the  Lelieuvre 
formulas  (§  79),  which  give  the  expressions  for  the  derivatives  of 
the  coordinates  of  S  in  terms  of  i/1?  i/2,  i>3. 

From  these  results  it  follows  that  any  three  solutions  of  an 

equation  of  the  form  ffQ 

=  MQ, 


where  M  is-  any  function  of  u  and  w,  determine  a  surface  S  upon 
which  the  parametric  curves  are  the  asymptotic  lines,  and  every 
other  solution  linearly  independent  of  these  three  gives  by  quad 
ratures  an  infinitesimal  deformation  of  S. 


EXAMPLES 

1.  A  necessary  and  sufficient  condition  that  two  surfaces  satisfying  the  condi 
tion  (2)  be  applicable  is  that  they  be  minimal  surfaces  adjoint  to  one  another. 

2.  If  x,  y,  z  and  xl9  T/I,  zi  satisfy  the  condition  (2),  so  also  do  £,  17,  f  and  &,  ^j, 
ft,  the  latter  being  given  by 

£  =  aix  +  biy  +  ciz  +  di,        xi  =  a^  +  a2Tn  +  asft  +  ei, 

77  =  «2«  +  &22/  +  C2Z  +  d2,  2/1  =  &l£l  +  &2^?l  +  &3ft  +  C2, 

f  =  a3x  -f  68y  +  c3z  +  d8,         Zi  =  Ci^  +  c2>?i  +  c3ft  +  c8, 
where  a1?  a2,  •  •  • ,  ei,  e2,  e3  are  constants. 

3.  A  necessary  condition  that  the  locus  of  the  point  (xi,  ?/i,  z\)  be  a  curve  is 
that  S  be  a  developable  surface.    In  this  case  any  orthogonal  trajectory  of  the 
tangent  planes  to  S  satisfies  the  condition. 

4.  Investigate  the  cases  0  =  0  and  0  =  c,  where  c  is  a  constant  different  from  zero. 

5.  If  Si  and  S{  correspond  to  S  with  orthogonality  of  linear  elements,  so  also 
does  the  locus  of  a  point  dividing  in  constant  ratio  the  line  joining  corresponding 
points  on  Si  and  S{. 


378  INFINITESIMAL  DEFORMATION 

155.  Associate  surfaces.  The  expressions  in  the  parentheses  of 
equation  (10)  differ  only  in  sign  from  the  second  fundamental  co 
efficients,  D0,  Z>J,  Z>0",  of  the  surface  /70  enveloped  by  the  plane 


(14) 

Hence  equation  (10)  may  be  written 

(15)  D"DQ  +  DD'J  -  2  D'D[  =  0. 

This  is  the  condition  that  to  the  asymptotic  lines  upon  either 
of  the  surfaces  S,  S0  there  corresponds  a  conjugate  system  on 
the  other  (§  56).  Bianchi  applies  the  term  associate  to  two  sur 
faces  whose  tangent  planes  at  corresponding  points  are  parallel, 
and  for  which  the  asymptotic  lines  on  either  correspond  to  a 
conjugate  system  on  the  other.  Since  the  converse  of  the  pre 
ceding  results  are  readily  shown  to  be  true,  we  have  the  theorem 
of  Bianchi  f  : 

When  two  surfaces  are  associate  the  expression  for  the  distance 
from  a  fixed  point  in  space  to  the  tangent  plane  to  one  is  the  char 
acteristic  function  for  an  infinitesimal  deformation  of  the  other. 

Hence  the  problems  of  infinitesimal  deformation  and  of  the 
determination  of  surfaces  associate  to  a  given  one  are  equivalent. 
We  consider  the  latter  problem. 

Since  the  tangent  planes  to  S  and  SQ  at  corresponding  points 
are  parallel,  we  have 

dzn      „  dx         dx          dxn         dx         dx 
-2  =  X  --  fji  —  ,  -5  =  0-  --  r  —  » 

du          du         dv  dv          du         du 

and  similar  equations  in  yQ  and  20,  where  X,  ft,  <r,  r  are  functions 
of  u  and  v  to  be  determined.  J 

If  these  equations  be  multiplied  by  —  »  —  »  --  and  added,  and 

~y        ~Y        2V-  $U          &M          dU 

likewise  by  —  »  —  »  —  •  and  added,  we  obtain 

dv     dv     dv 


\D<  - 

*  Cf.  §  67.  t  Lezioni,  Vol.  II,  p.  9. 

J  The  negative  signs  before  p.  and  r  are  taken  so  that  subsequent  results  may  have  a 
suitable  form. 


ASSOCIATE  SURFACES  379 

where  Z>0,  Z>0',  D0"  are  the  second  fundamental  quantities  for  $„. 
When  these  values  are  substituted  in  (15),  we  find 

(17)  X-r=0. 

Consequently  the  above  equations  reduce  to 

du          du         dv  dv          du         dv 

If  we  make  use  of  the  Gauss  equations  (V,  7),  the  condition  of 
integrability  of  equations  (18)  is  reducible  to 

du         dv 

where  A  and  B  are  determinate  functions.  Since  similar  equations 
hold  in  y  and  2,  both  A  and  B  must  be  identically  zero.  Calculating 
the  expressions  for  these  functions,  we  have  the  following  equations 
to  be  satisfied  by  X,  /A,  and  <r : 

JL    d\  .     f22i       .  rm  .     rii 


(19) 

da_d\         J22\      2     fl21          fill 

du      dv          v  1  J  I  1  J          I  1  J 

To  these  equations  we  must  add 
(20)  2  \D'-  pl>"—  <rZ>  =  0, 

obtained  from  the  last  of  (16).  The  determination  of  the  asso 
ciate  surfaces  of  a  given  surface  referred  to  any  parametric  system 
requires  the  integration  of  this  system  of  equations.  Moreover, 
every  set  of  solutions  leads  to  an  associate  surface.  We  shall  now 
consider  several  cases  in  which  the  parametric  curves  are  of  a 
particular  kind. 

156.  Particular  parametric  curves.  Suppose  that  S  is  a  sur 
face  upon  which  the  parametric  curves  form  a  conjugate  system. 
We  inquire  under  what  conditions  there  exists  an  associate  sur 
face  upon  which  also  the  corresponding  curves  form  a  conjugate 
system. 


380  INFINITESIMAL  DEFORMATION 

On  this  hypothesis  we  have,  from  (16), 

/*  =  a-  =  6, 
so  that  equations  (19)  reduce  to 


<»> 

which  are  consistent  only  when 


that  is,  when  the  point  equation  of  S,  namely 

j^    fi21<tf    ri2|<tf 

dudv     \l  J  cu      \ZJfo 

has  equal  invariants  (cf.  §  165). 

Conversely,  when  condition  (22)  is  satisfied,   the  function   X 
given  by  the  quadratures  (21)  makes  the  equations 


compatible,  and  thus  the  coordinates  of  an  associate  surface  are 
obtained  by  quadratures.  Hence  we  have  the  theorem  of  Cosserat*: 

The  infinitesimal  deformation  of  a  surface  S  is  the  same  problem 
as  the  determination  of  the  conjugate  systems  with  equal  point  invari 
ants  on  S. 

Since  the  relation  between  S  and  S0  is  reciprocal  and  the 
parametric  curves  are  conjugate  for  both  surfaces,  these  curves  on 
£0  also  have  equal  point  invariants. 

If  S  be  referred  to  its  asymptotic  lines,  the  corresponding  lines 
on  SQ  form  a  conjugate  system.  In  this  case,  as  is  seen  from  (16), 
X  is  zero  and  equations  (18)  reduce  to  . 

dxn  dx  dxn          dx  . 

/24>)  —  2  =  —  u  —  »  —  —  a-  —  ; 

du          ^dv  dv          3u 

moreover,  equations  (19)  become 

n 


Toulouse  Annales,  Vol.  VII  (1893),  N.  60. 


KULED  SUEFACES  381 

The  solution  of  this  system  is  the  same  problem  as  the  integra 
tion  of  a  partial  differential  equation  of  the  second  order,  as  is 
seen  by  the  elimination  of  either  unknown.  When  a  solution  of 
the  former  is  obtained,  the  corresponding  value  of  the  other 
unknown  is  given  directly  by  one  of  equations  (25). 

We  make  an  application  of  these  results  to  a  ruled  surface, 
which  we  suppose  to  be  referred  to  its  asymptotic  lines.  If  the 
curves  v  —  const,  are  the  generators,  they  are  geodesies,  and  conse 
quently  (VI,  50)  p1 

12 

Now  /*  can  be  found  by  a  quadrature.  When  this  value  is  sub 
stituted  in  the  second  of  equations  (25),  we  have  a  linear  equa 
tion  in  <r,  and  consequently  <r  also  can  be  obtained  by  quadratures. 
Hence  we  have  the  theorem : 

When  the  curved  asymptotic  lines  on  a  ruled  surface  are  known, 
its  associate  surfaces  can  be  found  by  quadratures. 

If  S0  were  referred  to  its  asymptotic  lines,  we  should  have 
equations  similar  to  (24).  These  equations  may  be  interpreted 
as  follows: 

The  tangent  to  an  asymptotic  line  on  one  of  two  associate  surfaces 
is  parallel  to  the  direction  conjugate  to  the  corresponding  curve  on 
the  other  surface. 

EXAMPLES 

1.  If  two  associate  surfaces  are  applicable  to  one  another,  they  are  minimal 
surfaces. 

2.  Every  surface  of  translation  admits  an  associate  surface  of  translation  such 
that  the  generatrices  of  the  two  surfaces  constitute  the  common  conjugate  system. 

3.  The  surfaces  associate  to  a  sphere  are  minimal. 

4.  When  the  equations  of  the  right  helicoid  are 

x  —  u  cos  v,         y  —  u  sin  u,         z  —  cro, 

the  characteristic  function  of  any  infinitesimal  deformation  is  0  =  ( U+  V)  (u2  +  «2)~  , 
where  U  and  V  are  arbitrary  functions  of  u  and  v  respectively.  Find  the  surfaces 
Si  and  So,  and  show  that  the  latter  are  molding  surfaces. 

5.  If  S0  and  S'0  are  associate  surfaces  of  a  surface  S,  the  locus  of  a  point 
dividing  in  constant  ratio  the  joins  of  corresponding  points  of  So  and  S6  is  an 
associate  of  S. 


382 


INFINITESIMAL  DEFOKMATION 


157.  Relations  between  three  surfaces  S,  S1?  S0.  Having  thus 
discussed  the  various  ways  in  which  the  problem  of  infinitesimal 
deformation  may  be  attacked,  we  proceed  to  the  consideration  of 
other  properties  which  are  possessed  by  a  set  of  three  surfaces 

$,          Stf         SQ. 

We  recall  the  differential  equation 

dxdxl  +  dydyv+  dzdzl  =  0, 

and  remark  that  it  may  be  replaced  by  the  three 
(26)       dXi=zQdy  —  yQdz,     dy^  xQdz  — zQdx,     dz^y^dx  — x^dy, 

if  the  functions  #0,  ?/0,  z0  are  such  a  form  that  the  conditions  of 
integrability  of  equations  (26)  are  satisfied.    These  conditions  are 


du  dv 

du  dv 

dx  dy, 
du  dv 


du  dv 

dx_d_z_ 
du  dv 


dv  du 

o  =  fe?5, 
dv  du 


dv  du 
d_x_dz, 
dv  du 
dy  dx( 


du  dv       dv  du       dv  du 


If  these  equations  be  multiplied  by  — »  — »  -  -   respectively   and 

added,  and  likewise  by  — i  — »  — ?  and  by  JT,  F,  ^,  we  obtain, 
by  (IV,  2),  "     0y     " 

(27) 


(28) 


From  the  first  two  of  these  equations  it  follows  that  the  locus  of 
the  point  with  coordinates  XQJ  T/O,  z0  corresponds  to  S  with  paral 
lelism  of  tangent  planes. 

In   order  to   interpret  the  last   of   these   equations   we   recall 
from  §  61  that 


-*-^ 

a% 

^4 

dv 

7 

A" 

r 

Z 

X 

Y 

Z  . 

^a; 

% 

dz 

a^ 

dy 

dz 

^it 

^?7 

du 

fin 

du 
~dv 

= 

fa 

du 

dv 
du 

dv 

du 

a  d(Y,  Z} 
ft  d(u,  v) 


Y= 


a  d(Z,  X) 
ft  d(u,  v) 


a  d(X,  Y) 
/if  d  (u,  v) 


KELATIONS  BETWEEN  S,  S19  AND  S0  383 

where  a  is  ±  1  according  as  the  curvature  of  the  surface  is  positive 
or  negative.    If  we  substitute  these  values  in  the  left-hand  mem 

bers  of  the  following  equations,  and  add  and  subtract  --- 

dx  dX  dX  dU   dv 

and  ---  from  these  equations  respectively,   the   resulting 

dv  du  dv 

expressions  are  reducible  to  the  form  of  the  right-hand  members 


du/         \      du 


dv         dv  \      du  dv 

By  means  of  these  and  similar  identities,  equation  (28)  can  be 
transformed  into 


,_  ^.  __D,^      *  Y      o  0 


Since  this  equation  is  equivalent  to  (15)  because  of  (27),  the 
quantities  a;0,  #0,  ZQ  in  (26)  are  the  coordinates  of  S0.  Hence 
when  a  surface  Sl  is  known,  the  coordinates  of  the  correspond 
ing  surface  S0  are  readily  found. 

This  result  enables  us  to  find  another  property  of  £0  and  Sr 
If  X^  Y^  Zt  denote  the  direction-cosines  of  the  normal  to  S^ 
they  are  given  by 

1 


I  d(u,  v)  l      HI  d(u,  v)  l      H^  d(u,  v) 


where  7/t  =  ^^—  F*,  E^  F^  Gl  being  the  coefficients  of  the 
linear  element  of  £r  If  the  values  of  the  derivatives  of  xv  yv,  zx, 
as  given  by  (26),  be  substituted  in  these  expressions,  we  have, 
in  consequence  of  (14), 

(30)  X^-^          Y 


As  an  immediate  consequence  we  have  the  theorem  : 

A  normal  to  St  is  parallel  to  the  radius  vector  of  S0  at  the  corre 
sponding  point. 


384  INFINITESIMAL  DEFORMATION 

By  means  of  (30)  we  find  readily  the  expressions  for  the  second 
fundamental  coefficients  J9t,  D[,  D'J  of  $L.    If  we  notice  that 


and  substitute  the  values  from  (6)  and  (30)  in 

du   du  1         ^  du    dv          ^4  dv    du 


we  obtain 


(31) 


From  these  expressions  follow 
(32) 


Combining  this  result  with  (15),  we  have  : 

The  asymptotic  lines  upon  any  one  of  a  group  of  three  surfaces 
S,  Sv  SQ  correspond  to  a  conjugate  system  on  the  other  two; 

or,  in  other  words: 

The  system  of  lines  which  is  conjugate  for  any  two  of  three  surfaces 
*Sf,  Sv  S0  corresponds  to  the  asymptotic  lines  on  the  other. 

If  the  curvature  of  S  be  negative,  its  asymptotic  lines  are  real, 
and  consequently  the  common  conjugate  system  on  S1  and  S0  is 
real.  If  these  lines  be  parametric,  the  second  of  equations  (32) 
reduces  to 


As  an  odd  number  of  the  four  quantities  in  this  equation  must 
be  negative,  either  SQ  or  S1  has  positive  curvature  and  the  other 
negative.  Similar  results  follow  if  we  begin  with  the  assumption 
that  Sl  or  SQ  has  negative  curvature. 

If  the  curvature  of  S  be  positive,  the  conjugate  system  common 
to  it  and  Sl  is  real  (cf.  §  56)  ;  consequently  the  asymptotic  lines 


RELATIONS  BETWEEN  S,  S»  AND  S0  385 

on  SQ  are  real,  and  the  curvature  of  the  latter  is  negative.  But 
we  saw  that  when  the  curvature  of  SQ  is  negative,  and  of  S  positive, 
that  of  S1  also  is  negative.  Hence  : 

Given  a  set  of  three  surfaces  S,  S^  SQ  ;  one  and  only  one  of  them 
has  positive  curvature. 

Suppose  that  S  is  referred  to  the  conjugate  system  corresponding 
to  asymptotic  lines  on  SQ.    The  point  equation  of  S  is 


We  shall  prove  that  this  is  the  point  equation  of  S^  also. 
If  we  differentiate  the  equation 


with  respect  to  v,  and  make  use  of  the  fact  that  y  and  z  are  solu 
tions  of  (33),  we  have,  in  consequence  of  (26), 

dudv  ~~  \dv  du~~l)vduj       I  1  /  #M      t  2  J~dv' 

But  the  expression  in  parenthesis  is  zero  in  consequence  of  equa 
tions  similar  to  (24),  and  hence  xl  is  a  solution  of  (33). 

Since  the  parametric  curves  on  S0  are  its  asymptotic  lines,  the 
spherical  representation  of  £0  and  consequently  of  S  must  satisfy 
the  condition  ^  f!2V  d  fl2V 

Hence  we  have  the  theorem  of  Cosserat: 

The  problem  of  infinitesimal  deformation  of  a  surface  is  the  same 
as  the  determination  of  the  conjugate  systems  with  equal  tangential 
invariants  upon  the  surface. 

158.  Surfaces  resulting  from  an  infinitesimal  deformation.    We 

pass  to  the  consideration  of  the  surface  S'  arising  from  an  infini 
tesimal  deformation  of  8.  Its  coordinates  are  given  by 

where  €  is  a  small  constant  whose  powers  higher  than  the  first  are 
neglected.  Since  the  fundamental  quantities  of  the  first  order  for 
$',  namely J£"',  Ff,  G',  are  equal  to  the  corresponding  ones  for  £,  by 


386  INFINITESIMAL  DEFORMATION 

means  of  (26)  the  expressions  for  the  direction-cosines  X1,  F',  Z' 
of  the  normal  to  S'  are  reducible  to 

and  similar  expressions  for  Y'  and  Z'. 

The  derivatives  of  X'  with  respect  to  u  and  v  are  reducible  by 
means  of  (29)  to 

dX^_d_X       /    dY_      dZ\      ea  /D,^_D  3X' 

dX          dX            I       OJL                vZ\        €d   I  _  .1  uX             i  vJL 
i     fly 77    I    I     - (    T) 7)   

dv        dv         \  °  dv  dv  /      /i-  \    °  du  dv 

where  a  is  ±1  according  as  the  curvature  of  $0  is  positive  or  negative. 
When  these  results  are  combined  with  (26)  and  (34),  we  obtain 

^^_V  — —      —  V  —  (D'  —  -D  — 
du  du      ^4  du  du      /if  ^  du  \    °  du  dv 


e+-v 

**l'\9*'9*      du  cu)     *°\du  du      du  du 

The  last  expression  is  identically  zero,  as  one  sees  by  writing  it 

fix1  flX9 

out  in  full.     From   this   and  similar  expressions  for  V-        — » 
a- aTl  ,  ,«v/  A^  ay 

X  — — ,  and  V  —  — ,  the  values  for  the  second  fundamental 
dv   du  **  dv   dv 

coefficients  of  S'  can  be  given  in  the  form 


= -  2)  T  T- =D + jf 

**   u  cu 


(30) 


We  know  that  ff  is  equal  to  ±  UK  according  as  the  curvature 
of  S  is  positive  or  negative  (cf.  §  60).  Also,  by  §  157,  one  and  only 
one  of  three  surfaces  S,  Sv  S0  has  positive  curvature.  Recalling 
that  a  in  the  above  formulas  is  ±  1  according  as  the  curvature  of 
S0  is  positive  or  negative,  we  can,  in  consequence  of  (31),  write 
equations  (36)  in  the  form 

Z>       .*"  =  .ZX'  ±         J>», 


where  the  upper  sign  holds  when  Sl  has  positive  curvature. 


ISOTHERMIC  SUBFACES  387 

From  these  equations  it  is  seen  that  &'  and  Dr  can  be  zero  sim 
ultaneously  only  when  D[  is  zero.  Hence  we  have: 

The  unique  conjugate  system  which  remains  conjugate  in  an  infini 
tesimal  deformation  of  a  surface  is  the  one  corresponding  to  a  conju 
gate  system  on  S^,  or,  what  is  the  same  thing,  to  the  asymptotic  lines 
on  SQ. 

In  particular,  in  order  that  the  curves  of  this  conjugate  system 
be  the  lines  of  curvature,  it  is  necessary  and  sufficient  that  the 
spherical  representation  be  orthogonal,  and  consequently  that  £0  be 
a  minimal  surface  (cf.  §  55).  From  this  it  follows  that  the  spherical 
representation  of  the  lines  of  curvature  of  S  is  isothermal.  Con 
versely,  if  a  surface  is  of  this  kind,  there  is  a  unique  minimal  sur 
face  with  the  same  representation  of  its  asymptotic  lines,  and  this 
surface  can  be  found  by  quadratures.  Hence  the  required  infinites 
imal  deformation  of  the  given  surface  can  be  effected  by  quadra 
tures  (26),  and  so  we  have  the  theorem  of  Weingarten  *  : 

A  necessary  and  sufficient  condition  that  a  surface  admit  an  infini 
tesimal  deformation  which  preserves  its  lines  of  curvature  is  that  the 
spherical  representation  of  the  latter  be  isothermal;  when  such  a 
surface  is  expressed  in  terms  of  parameters  referring  to  its  lines  of 
curvature,  the  deformation  can  be  effected  by  quadratures. 

159.  Isothermic  surfaces.  By  means  of  the  results  of  §  158  we 
obtain  an  important  theorem  concerning  surfaces  whose  lines  of 
curvature  form  an  isothermal  system.  They  are  called  isothermic 
surfaces  (cf.  Exs.  1,  3,  p.  159). 

From  equations  (23)  it  follows  that  if  the  common  conjugate 
system  on  two  associate  surfaces  is  orthogonal  for  one  it  is  the 
same  for  the  other.  In  this  case  equation  (22)  reduces  to 


of  which  the  general  integral  is 

E      U 

G=r 

where  U  and  V  are  functions  of  u  and  v  respectively.    Hence  the 
lines  of  curvature  on  S  form  an  isothermal  system  (cf.  §  41). 

*  Sitzungsberichte  der  Konig.  Akademie  zu  Berlin,  1886. 


388  INFINITESIMAL  DEFORMATION 

If  the  parameters  be  isothermic  and  the  linear  element  written 

ds2=r(du2+dv2), 
it  follows  from  (21)  that 

(37)  X  =  i, 
and  equations  (23)  become 

du  ~~  r  du'          dv  ~       r  dv' 

From  these  results  we  derive  the  following  theorem  of  Bour  *  and 
Christoffel : 

If  the  linear  element  of  an  isothermic  surface  referred  to  its  lines 
of  curvature  be  ds*  _  r  /du'*  _|_  dl?\ 

a  second  isothermic  surface  can  be  found  by  quadratures.  It  is  asso 
ciate  to  the  given  one,  and  its  linear  element  is 

1 

ds?  —  —  (du  +  dv  ). 
r 

From  equations  (16)  and  (17)  it  follows  that  the  equation  of  the 
common  conjugate  system  (IV,  43)  on  two  associate  surfaces  $,  S0 
is  reducible  to 

(38)  fi  du2  +  2  X  dudv  +  a  dv2  =  0. 

The  preceding  results  tell  us  that  a  necessary  and  sufficient  condi 
tion  that  S  be  an  isothermic  surface  is  that  there  be  a  set  of  solu 
tions  of  equations  (19)  such  that  (38)  is  the  equation  of  the  lines 
of  curvature  on  S.  Hence  there  must  be  a  function  p  such  that 

p  —  p  (ED*  —  FD),     2  X  ==  p  (ED1'  —  GD),     <r  =?<p  (FD"  —  GDf) 
satisfy  equations  (19).f    Upon  substitution  we  are  brought  to  two 
equations  of  the  form 

S-s-  =  a;,          —-  =  p\ 

du  dv 

where  a  and  ft  are  determinate  functions  of  u  and  v.  In  order  that 
S  be  isothermic,  these  functions  must  satisfy  the  condition 

dv  ~~  du ' 

When  it  is  satisfied,  p  and  consequently  p,  X,  a  are  given  by  quad 
ratures. 

*  Journal  de  I'Ecole  Poly  technique,  Cahier  39  (18G2),  p.  118.     / 
f  Cf.  Bianchi,  Vol.  II,  p.  30. 


ISOTHEKMIC  SURFACES  389 

Consider  furthermore  the  form 
(39)  H(p  du*  +  2  X  dudv  +  o-  dv~). 

From  (37)  it  is  seen  that  when  the  lines  of  curvature  are  para 
metric,  this  expression  reduces  to  2  dudv.  Hence  its  curvature  is 
zero  (cf.  V,  12),  and  consequently  the  curvature  of  (39)  is  zero. 
From  §  135  it  follows  that  this  form  is  reducible  to  du1dvl  by  quad 
ratures.  Hence  we  have  the  theorem  of  Weingarten : 

The  lines  of  curvature  upon  an  isothermic  surface  can  be  found  by 
quadratures. 

We  conclude  this  discussion  of  isothermic  surfaces  with  the  proof  of  a  theorem 
of  Ribaucour.  He  introduced  the  term  limit  surfaces  of  a  group  of  applicable  sur 
faces  to  designate  the  members  of  the  group  whose  mean  curvature  is  a  maximum 
or  minimum.  According  to  Ribaucour, 

The  limit  surfaces  of  a  group  of  applicable  surfaces  are  isothermic. 

In  proving  it  we  consider  a  member  S  of  the  group  referred  to  its  lines  of  cur 
vature.  Its  mean  curvature  is  given  by  D/E  +  D"/G.  In  consequence  of  equations 
(36)  the  mean  curvature  of  a  near-by  surface  is,  to  within  terms  of  higher  order, 


A  necessary  and  sufficient  condition  that  the  mean  curvature  of  S  be  a  maximum 
or  minimum  is  consequently  /j)      j>"\ 


Excluding  the  case  of  the  sphere  for  which  the  expression  in  parenthesis  is  zero, 
we  have  that  DO  is  zero.  Hence  the  common  conjugate  system  of  S  and  <S0  is  com 
posed  of  lines  of  curvature  on  the  former,  and  therefore  S  is  isothermic. 

GENERAL  EXAMPLES 

1.  If  x,  y,  «  and  xi,  y\,  z\  are  the  coordinates  of  two  surfaces  corresponding  with 
orthogonality  of  linear  elements,  the  coordinates  of  a  pair  of  applicable  surfaces 
are  given  by  fc  =  x  +  tei,        m  =  y  +  ty\,        n  =  «  +  <zi, 

£2  =  x  —  txi,         -r}2  —  y  —  tyi,         f2  =  z  —  tei, 
where  t  is  any  constant. 

2.  If  two  surfaces  are  applicable,  the  locus  of  the  mid-point  of  the  line  joining 
corresponding  points  admits  of  an  infinitesimal  deformation  in  which  this  line  is 
the  generatrix. 

3.  Whatever  be  the  surface  S,  the  characteristic  equation  (7)  admits  the  solu 
tion  0  =  aX  +  bY  +  cZ,  where  a,  6,  c  are  constants.    Show  that  S0  is  the  point 
(a,  6,  c)  and  that  equations  (26)  become 

xi  =  cy  -  bz  +  d,        yi  =  az  —  ex  -\-  e,        z\  =  bx  —  ay  +  /, 

where  d,  e,  /are  constants;  that  consequently  Si  is  a  plane,  and  that  the  infinitesi 
mal  deformation  is  in  reality  an  infinitesimal  displacement. 


390  INFINITESIMAL  DEFORMATION 

4.  Determine  the  form  of  the  results  of  Exs.  1,  2,  where  0  has  the  value  of  Ex.  3. 

5.  Show  that  the  first  fundamental  coefficients  EI,  FI,  GI  of  a  surface  Si  are  of 
the  form 


, 

E1  =  E<f>*      - 


=  -F02  ,  _.- 

dv 


6.  Let  S  denote  the  locus  of  the  point  which  bisects  the  segment  of  the  normal 
to  a  surface  S  between  the  centers  of  principal  curvature  of  the  latter.    In  order 
that  the  lines  on  2  corresponding  to  the  lines  of  curvature  on  S  shall  form  a  conju 
gate  system,  it  is  necessary  and  sufficient  that  S  correspond  to  a  minimal  surface 
with  orthogonality  of  linear  elements,  and  that  the  latter  surface  and  S  correspond 
with  parallelism  of  tangent  planes. 

7.  Show  that  when  the  spherical  representation  of  the  asymptotic  lines  of  a  sur 
face  S  satisfies  the  condition  \\y      a  (92 


cu  (  2  }       cv  (  1 

equations  (25)  admit  two  pairs  of  solutions  which  are  such  that  /x  =  <r  and  /j.  =  —  <r. 
On  the  two  associate  surfaces  S0,  SQ  thus  found  by  quadratures  the  parametric 
systems  are  isothermal-conjugate,  and  SQ  and  SQ  are  associates  of  one  another. 

8.  Show  that  the  equation  of  Ex.  7  is  a  necessary  and  sufficient  condition  that 
two  surfaces  associate  to  S  be  associate  to  one  another. 

9.  Show  that  when  the  sphere  is  referred  to  its  minimal  lines,  the  condition  of 
Ex.  7  is  satisfied,  and  investigate  this  case. 

10.  On  any  surface  associate  to  a  pseudospherical  surface  the  curves  correspond 
ing  to  the  asymptotic  lines  of  the  latter  are  geodesies.    A  surface  with  a  conjugate 
system  of  geodesies  is  called  a  surface  of  Voss  (cf.  §  170). 

11.  Determine  whether  minimal  surfaces  and  the  surfaces  associate  to  pseudo- 
spherical  surfaces  are  the  only  surfaces  of  Voss. 

12.  When  the  equations  of  a  central  quadric  are  in  the  form  (VII,  35),  the  asso 
ciate  surfaces  are  given  by 


2/o  =  2  V6  Fj  Uu  du  +  f  Vv  dv\  , 


z0  =  i 

where  U  and  V  are  arbitrary  functions  of  u  and  v  respectively  ;  hence  the  associates 
are  surfaces  of  translation. 

13.  When  the  equations  of  a  paraboloid  are  in  the  form 

x=Va(u  +  1>),    y=Vb(u-v),    z  =  2uv, 

the  associate  surfaces  are  surfaces  of  translation  whose  generators  are  plane  curves  ; 
their  equations  are 

x0  =  Va(U  +  V),     y0=Vb(V-U),     z0  =  2fuU'du 
where  U  and  V  are  arbitrary  functions  of  u  and  v  respectively. 


GENERAL  EXAMPLES  391 

14.  Show  that  a  quadric  admits  of  an  infinitesimal  deformation  which  preserves 
its  lines  of  curvature,  and  determine  the  corresponding  associate  surface. 

15.  Since  the  relation  between  S  arid  Si  is  reciprocal,  there  is  a  surface  <S3 
associate  to  Si  which  bears  to  S  a  relation  similar  to  that  of  SQ  to  Si.    Show  that 
the  asymptotic  lines  on  S0  and  S3  correspond,  and  that  these  surfaces  are  polar 
reciprocal  with  respect  to  the  imaginary  sphere  z2  +  ?/2  +  z2  +  1  =  0. 

16.  Since  the  relation  between  S  and  So  is  reciprocal,  there  is  a  surface  S%  cor 
responding  to  S0  with  orthogonality  of  linear  elements  which  bears  to  S  a  relation 
similar  to  that  of  Si  to  So.    Show  that  the  asymptotic  lines  on  Si  and  Sz  correspond, 
that  the  coordinates  of  the  latter  are  such  that 

xi-xz  =  yzo  -  zy0,    yi-yz  =  zx0  -  zz0,     z\  —  z*  =  xy0  -  yx0, 
and  that  the  line  joining  corresponding  points  on  Si  and  S2  is  tangent  to  both  surfaces. 

17.  Show  that  if  S5  denotes  the  surface  corresponding  to  S3  with  orthogonality 
of  linear  elements  which  is  determined  by  Si,  associate  to  SB,  the  surfaces  S  and 
S&  are  related  to  one  another  in  a  manner  similar  to  Si  and  Sz  of  Ex.  16. 

18.  Show  that  the  surface  S4,  which  is  the  associate  to  Sz  determined  by  So,  is 
the  polar  reciprocal  of  <S  with  respect  to  the  imaginary  sphere  x'2  +  y2  +  z2  +  1  =  0. 

19.  If  we  continue  the  process  introduced  in  the  foregoing  examples,  we  obtain 
two  sequences  of  surfaces 

S,       Si,       $3,       Sj,       S7,       Sg,       Sn,       •  -  •  , 

S,     So,     S2,     84,     Se,     $8,     Sio, 

Show  that  Sn  and  S10  are  the  same  surface,  likewise  Si2  and  S9,  and  that  conse 
quently  there  is  a  closed  system  of  twelve  surfaces ;  they  are  called  the  twelve  sur 
faces  of  Darboux. 

20.  A  necessary  and  sufficient  condition  that  a  surface  referred  to  its  minimal 
lines  be  isothermic  is  that  j)       jj 

D"     F' 

where  U  and  V  are  functions  of  u  and  v  respectively. 

21.  A  necessary  and  sufficient  condition  that  the  lines  of  curvature  on  an  iso 
thermic  surface  be  represented  on  the  sphere  by  an  isothermal  system  is  that 

Pi_U 

P*  ~  F' 

where  U  and  V  are  functions  of  u  and  v  respectively,  the  latter  being  parameters 
referring  to  the  lines  of  curvature.  Show  that  the  parameters  of  the  asymptotic 
lines  on  such  a  surface  can  be  so  chosen  that  E  =  G. 

22.  Show  that  an  isothermic  surface  is  transformed  by  an  inversion  into  an 
isothermic  surface. 

23.  If  Si  and  S2  are  the  sheets  of  the  envelope  of  a  family  of  spheres  of  two 
parameters,  which  are  not  orthogonal  to  a  fixed  sphere,  and  the  points  of  contact  of 
any  sphere  are  said  to  correspond,  in  order  that  the  correspondence  be  conformal, 
it  is  necessary  that  the  lines  of  curvature  on  Si  and  S2  correspond  and  that  these 
surfaces  be  isothermic  (cf.  Ex.  15,  Chap.  XIII). 


CHAPTER  XII 

RECTILINEAR  CONGRUENCES 

160.  Definition  of  a  congruence.  Spherical  representation.  A  two- 
parameter  system  of  straight  lines  in  space  is  called  a  rectilinear 
congruence.  The  normals  to  a  surface  constitute  such  a  system ; 
likewise  the  generatrices  of  an  infinitesimal  deformation  of  a  sur 
face  (cf.  §  152).  Later  we  shall  find  that  in  general  the  lines  of  a 
congruence  are  not  normal  to  a  surface.  Hence  congruences  of 
normals  form  a  special  class ;  they  are  called  normal  congruences. 
They  were  the  first  studied,  particularly  in  investigations  of  the 
effects  of  reflection  and  refraction  upon  rays  of  light.  The  first 
purely  mathematical  treatment  of  general  rectilinear  congruences 
was  given  by  Kummer  in  his  memoir,  Allgemeine  Theorie  der 
gradlinigen  Strahlensysteme.*  We  begin  our  treatment  of 
the  subject  with  the  derivation  of  certain  of  Rummer's  results  by 
methods  similar  to  his  own. 

From  the  definition  of  a  congruence  it  follows  that  its  lines 
meet  a  given  plane  in  such  a  way  that  through  a  point  of  the 
plane  one  line,  or  at  most  a  finite  number,  pass.  Similar  results 
hold  if  a  surface  be  taken  instead  of  a  plane ;  this  surface  is 
called  the  surface  of  reference.  And  so  we  .rrnay  define  a  con 
gruence  analytically  by  means  of  the  coordinates  of  the  latter 
surface  in  terms  of  two  parameters  u,  v,  and  by  the  direction- 
cosines  of  the  lines  in  terms  of  these  parameters.  Thus,  a  con 
gruence  is  defined  by  a  set  of  equations  such  as 

*—f\(uiv}'>     y  =fz(u->  v)->     z—jz(u->v)'> 

where  the  functions  /  and  <£  are  analytic  in  the  domain  of  u  and  v 
under  consideration,  and  the  functions  (/>  are  such  that 


*  Crelle,  Vol.  LVII  (1860),  pp.  189-230. 
302 


NORMAL  CONGRUENCES  393 

We  make  a  representation  of  the  congruence  upon  the  unit 
sphere  by  drawing  radii  parallel  to  the  lines  of  the  congruence,  and 
call  it  the  spherical  representation  of  the  congruence.  When  We  put 


the  linear  element  of  the  spherical  representation  is 
(3)  da2  =  <f(^2  +  2  &dudv 

If  we  put 

^A  dx  dX       ,,          ex  dX  dx  dX  dx 

=' 


we  have  the  second  quadratic  form 

(5)  ]£  dxdX=  e  du2  +  (/  +/')  rfwdv  +  g  dv\ 
which  is  fundamental  in  the  theory  of  congruences. 

161.  Normal  congruences.  Ruled  surfaces  of  a  congruence.  If 
there  be  a  surface  S'  normal  to  the  congruence,  the  coordinates 
of  S'  are  given  by 

(6)  x'=x  +  tX,         y'=y  +  tY,          z'=z  +  tZ, 

where  t  measures  the  distance  from  the  surface  of  reference  to  Sr. 
Since  £'  is  normal  to  the  congruence,  we  must  have 

(7) 

which  is  equivalent  to 


du     du  dv      3v 

If  these  equations  be  differentiated  with  respect  to  v  and  u  respec 
tively,  and  the  resulting  equations  be  subtracted,  we  obtain 

(9)  /=/'. 

Conversely,  when  this  condition  is  satisfied,  the  function  t  given 
by  the  quadratures  (8)  satisfies  equation  (7).  Since  t  involves  an 
additive  constant,  equations  (6)  define  a  family  of  parallel  surfaces 
normal  to  the  congruence.  Hence  : 

A  necessary  and  sufficient  condition  for  a  normal  congruence  is 
that  f  and  f  be  equal. 

The  lines  of  the  congruence  which  pass  through  a  curve  on  the 
surface  of  reference  S  form  a  ruled  surface.    Such  a  curve,  and 


394  RECTILINEAR  CONGRUENCES 

consequently  a  ruled  surface  of  the  congruence,  is  determined  by  a 
relation  between  u  and  v.  Hence  a  differential  equation  of  the  form 

(10)  Mdu+Ndv  =  0 

defines  a  family  of  ruled  surfaces  of  the  congruence.  We  consider 
a  line  l(u,  v)  of  the  congruence  and  the  ruled  surface  2  of  this 
family  upon  which  I  is  a  generator  ;  we  say  that  2  passes  through  I. 
We  apply  to  2  the  results  of  §§  103,  104. 

If  dsQ  denotes  the  linear  element  of  the  curve  C  in  which  2  cuts 
the  surface  of  reference,  it  follows  from  (VII,  54),  (3),  and  (5)  that 
the  quantities  a2  and  b  for  2  have  the  values 

\*     da2 


^  dX  dx 
~ 


From  (VII,  58)  we  have  that  the  direction-cosi'ies  X,  //.,  v  of  the 
common  perpendicular  to  I  and  to  the  line  I'  of  parameters  u  4-  du, 
v  +  dv,  where  dv/du  is  given  by  (10),  have  the  values 


(12) 

-  \     da          da 

which,  by  means  of  (V,  31),  are  reducible  to 

dX      ^dX\  ,     ,  /  ^dX      *>dX\-, 
—  -  &—  )du  +  {  <?—-  -  3—    dv 
dv  du  /  \     dv  du/ 

(13)  \= 


&*  .  da 

and  similar  expressions  for  /A  and  v. 
From  (12)  it  follows  that 

.dX  ,       dY  t      dZ      n 

X^~  +  ^^~"hz/^~=0- 
da          da         da 

Since  dX/da,  dY/da,  dZ/da  are  the  direction-cosines  of  the  tangent 
to  the  spherical  representation  of  the  generators  of  2,  we  have  the 
theorem  : 

Given  a  ruled  surface  2  of  a  congruence  ;  let  C  be  the  curve  on 
the  unit  sphere  which  represents  2,  and  M  the  point  of  C  correspond 
ing  to  a  generator  L  of  S  ;  the  limiting  position  of  the  common  per 
pendicular  to  L  and  a  near-by  generator  of  2  is  perpendicular  to  the 
tangent  to  C  at  M. 


PRINCIPAL  SURFACES 


395 


162.  Limit  points.  Principal  surfaces.  By  means  of  (VII,  62) 
and  (12)  we  find  that  the  expression  for  the  shortest  distance  8 
between  I  and  V  is,  to  within  terms  of  higher  order, 

dx      dy      dz 


dsf 
da- 


X 
dX 


Y 
dY 

dS« 


z 

dZ 


When  the  values  (13)  for  X,  /*,  v  are  substituted  in  the  right-hand 
member  of  this  equation,  the  result  is  reducible  to 


(14) 


/{do- 


<odu 
e  du 


+  gdv 
'du  +  g  dv 


If  jY  denotes  the  point  where  this  line  of  shortest  distance  meets  £, 
the  locus  of  jVis  the  line  of  striction  of  2.  Hence  the  distance  of 
N  from  the  surface  $,  measured  along  Z,  is  given  by  (VII,  65) ;  if 
it  be  denoted  by  r,  we  have,  from  (11), 

n  .  _  edu*+(f+f)  dudv  +  g  dv2 

V10/  r  jC-J«.2    i 


For  the  present*  we  exclude  the  case  where  the  coefficients  of 
the  two  quadratic  forms  are  proportional.  Hence  r  varies  with 
the  value  of  dv/du,  that  is,  with  the  ruled  surface  2  through  I.  If 
we  limit  our  consideration  to  real  surfaces  2,  the  denominator  is 
always  positive,  and  consequently  the  quantity  r  has  a  finite  maxi 
mum  and  minimum.  In  order  to  find  the  surfaces  2  for  which  r 
has  these  limiting  values,  we  replace  dv/du  by  £,  and  obtain 


(16) 

If  we  equate  to  zero  the  derivative  of  the  right-hand  member  with 
respect  to  £,  we  get 

a  quadratic  in  t.    Since  €$—&*>  0,  we  may  apply  to  this  equation 
reasoning  similar  to  that  used  in  connection  with  equation  (IV,  21), 

*  Cf.  Ex.  1,  §  171. 


396 


RECTILINEAR  CONGRUENCES 


and  thus  prove  that  it  has  two  real  roots.  The  corresponding  values 
of  r  follow  from  (16)  when  these  values  of  t  are  substituted  in  the 
latter.  Because  of  (17)  the  resulting  equation  may  be  written 


r  —  — 


where  t  indicates  a  root  of  (17)  and  r  the  corresponding  value  of  r. 
When  we  write  the  preceding  equations  in  the  form 


\€  r  +  e]  +  [$r  +  \ 


-o, 

and  eliminate  t,  we  obtain  the  following  quadratic  in  r: 

If  r^  and  r2  denote  the  roots  of  this  equation,  we  have 
(19) 


The  points  on  I  corresponding  to  these  values  of  r  are  called  its 
limit  points.  They  are  the  boundaries  of  the  segment  of  I  upon 
which  lie  the  feet  of  each  perpendicular  common  to  it  and  to  a 
near-by  line  of  the  congruence.  The  ruled  surfaces  of  the  con-, 
gruences  which  pass  through  I  and  are  determined  by  equation 
(17)  are  called  the  principal  surfaces  for  the  line.  There  are  two 
of  them,  and  their  tangent  planes  at  the  limit  points  are  determined 
by  I  and  by  the  perpendiculars  of  shortest  distance  at  the  limit 
points.  They  are  called  the  principal  planes. 

In  order  to  find  other  properties  of  the  principal  surfaces,  we 
imagine  that  the  parametric  curves  upon  the  sphere  represent  these 
surfaces.  If  equation  (17)  be  written 

€du 

iu  +  l 


(20) 


=  0, 


PRINCIPAL  SURFACES  397 

it  is  seen  that  a  necessary  and  sufficient  condition  that  the  ruled 
surfaces  v  =  const.,  u  =  const,  be  the  principal  surfaces,  is 


From  thes"e  it  follows  that  since  the  coefficients  of  the  two  funda 
mental  quadratic  forms  are  not  proportional,  we  must  have 

(21)  ^=0,        /+/'=0. 

From  the  first  of  these  equations  and  the  preceding  theorem  follows 
the  result: 

The  principal  surfaces  of  a  congruence  are  represented  on  the 
sphere  by  an  orthogonal  system,  and  the  two  principal  planes  for 
each  line  are  perpendicular  to  one  another. 

For  this  particular  parametric  system  equation  (13)  reduces  to 

^9X,        ^dX. 
<o  —  du  —  &  —  dv 


(22) 


so  that  the  direction-cosines  Xx,  JJL^  vl  of  the  perpendicular  whose  foot 
is  the  limit  point  on  I  corresponding  to  v  =  const,  have  the  values 

1  ay  l   dz 


Hence  the  angle  GO  between  the  lines  with  these  direction-cosines 

and  those  with  (22)  is  given  by 

.  7 
du 


cos  &>  = 


The  values  of  r1  and  r2  are  now 

e  a 

ri=--r«      r^~S' 

©  ^ 

so  that  with  the  aid  of  (23)  equation  (15)  can  be  put  in  the  form 
(24)  r  =  rl  cos2  &>  -f  r2  sin2  co. 

This  is  Hamilton's  equation.    We  remark  that  it  is  independent  of 
the  choice  of  parameters. 


398  KECTILINEAR  CONGEUENCES 

163.  Developable  surfaces  of  a  congruence.  Focal  surfaces.  In 
order  that  a  ruled  surface  be  developable,  it  is  necessary  and  suffi 
cient  that  the  perpendicular  distance  between  very  near  generators 
be  of  the  second  or  higher  order.  From  (14)  it  follows  that  the 
ruled  surfaces  of  a  congruence  satisfying  the  condition 


(25) 


0 


e  du  -}-fdv,          f'du  +  g  dv 

are  developable.  Unlike  equation  (20),  the  values  of  dv/du  satis 
fying  this  equation  are  not  necessarily  real.  We  have  then  the 
theorem  : 

Of  all  the  ruled  surfaces  of  a  congruence  through  a  line  of  it  two 
are  developable,  but  they  are  not  necessarily  real. 

The  normals  to  a  real  surface  afford  an  example  of  a  congruence 
with  real  developables  ;  for,  the  normals  along  a  line  of  curvature 
form  a  developable  surface  (§  51).  Since  /and/'  are  equal  in  this 
case,  equations  (20)  and  (25)  are  equivalent.  And,  conversely,  they 
are  equivalent  only  in  this  case.  Hence  : 

When  a  congruence  is  normal,  and  only  then,  the  principal  surfaces 
are  developable. 

When  a  ruled  surface  is  developable  its  generators  are  tangent 
to  a  curve  at  the  points  where  the  lines  of  shortest  distance  meet 
them.  Hence  each  line  of  a  congruence  is  tangent  to  two  curves 
in  space,  real  or  imaginary  according  to  the  character  of  the  roots 
of  equation  (25).  The  points  of  contact  are  called  the  focal  points 
for  the  line.  By  means  of  (25)  we  find  that  the  values  of  r  for 
these  points  are  given  by 

e  du  -\-fdv  _       f'du+g  dv 


If  these  equations  be  written  in  the  form 

(£p  4-  e)du  +  (&p  +f)dv  =  0, 
(<&!>  +/')  du  4-  (gp  +g}dv  =  0, 

and  if  du,  dv  be  eliminated,  we  have 

(26) 


DEVELOPABLE  SURFACES  399 

If  pl  and  p2  denote  the  roots  of  this  equation,  it  follows  that 


(27) 


A=^-//' 


From  (19)  and  (27)  it  is  seen  that 
(28) 


These  results  may  be  interpreted  as  follows  : 

The  mid-points  of  the  two  segments  bounded  respectively  by  the 
limit  points  and  by  the  focal  points  coincide. 

This  point  is  called  the  middle  point  of  the  line  and  its  locus  the 
middle  surface  of  the  congruence. 

The  distance  between  the  focal  points  is  never  greater  than  that 
between  the  limit  points.    They  coincide  when  the  congruence  is  normal. 

Equation  (24)  may  be  written  in  the  forms 

cos2  &)  =  -  -  1         sin2  &)  =  —  -- 
r  —  r  r  —  r 

'i      '2  ri      ri 

Hence  if  a)1  and  &)2  denote  the  values  of  &)  corresponding  to  the 
developable  surfaces,  we  have 


A*       T      ^,     A* 


From  these  and  the  first  of  (28)  it  follows  that 

cos2ft)1  =  sin2&)2,         sin2ft)1  =  cos2ft)2, 
so  that 

(29)  cos2&)1+cos2ft)2=0, 
and  consequently 

(30)  w1+o)2=|±ww, 
jor 

(31)  ft)1~ft)2= 


400  RECTILINEAR  CONGRUENCES 

where  n  denotes  any  integer.  If  the  latter  equation  be  true,  the 
developable  surfaces  are  represented  on  the  sphere  by  an  orthog 
onal  system,  as  follows  from  the  theorem  at  the  close  of  §  161.  But 
by  §  34  the  condition  that  equation  (25)  define  an  orthogonal  sys 
tem  on  the  sphere  is/=/',  that  is,  the  congruence  must  be  normal. 
Since  in  this  case  the  principal  surfaces  are  the  developables,  equa 
tion  (30)  as  well  as  (31)  is  satisfied.  Hence  equation  (30)  is  the 
general  solution  of  (29). 

The  planes  through  I  which  make  the  angles  o^,  &>2  with  the 
principal  plane  w  =  0  are  called  the  focal  planes  for  the  line  ;  they 
are  the  tangent  planes  to  the  two  developable  surfaces  through 
the  line.  Incidentally  we  have  proved  the  theorem: 

A  necessary  and  sufficient  condition  that  the  two  focal  planes  for 
each  line  of  a  congruence  be  perpendicular  is  that  the  congruence 
be  normal. 

And  from  equation  (30)  it  follows  that 

The  focal  planes  are  symmetrically  placed  with  respect  to  the  prin 
cipal  planes  in  such  a  way  that  the  angles  formed  by  the  two  2iairs 
of  planes  have  the  same  bisecting  planes. 

If  6  denote  the  angle  between  the  focal  planes,  then 

and  ^ 

(32)  sin  6  —  cos  2  a)l—  cos2o)1—  cos2o>2  =  - l — —  • 

The  loci  of  the  focal  points  of  a  congruence  are  called  its  focal 
surfaces.  Each  line  of  the  congruence  touches  both  surfaces,  being 
tangent  to  the  edges  of  regression  of  the  two  developables  through  it. 
By  reasoning  similar  to  that  employed  in  the  discussion  of  surfaces 
of  center  (§  74)  we  prove  the  theorem : 

A  congruence  may  be  regarded  as  two  families  of  developable  sur 
faces.  Each  focal  surface  is  touched  by  the  developables  of  one  family 
along  their  edges  of  regression  and  enveloped  by  those  of  the  other 
family  along  the  curves  conjugate  to  these  edges. 

The  preceding  theorem  shows  that  of  the  two  focal  planes  through 
a  line  I  one  is  tangent  to  the  focal  surface  SL  and  the  other  is  the 


ASSOCIATE  NORMAL  CONGRUENCES  401 

osculating  plane  of  the  edge  of  regression  on  /S\  to  which  I  is  tan 
gent  ;  similar  results  hold  for  Sz.  When  the  congruence  is  nor 
mal  these  planes  are  perpendicular,  and  consequently  these  edges 
of  regression  are  geodesies  on  Sl  and  Sz.  Since  the  converse  is 
true  (§  76),  we  have: 

A  necessary  and  sufficient  condition  that  the  tangents  to  a  family 
of  curves  on  a  surface  form  a  normal  congruence  is  that  the  curves  be 
geodesies. 

EXAMPLES 

1.  If  JT,  Y",  Z  are  the  direction-cosines  of  the  normal  to  a  minimal  surface  at 
the  point  (cc,  T/,  z),  the  line  whose  direction-cosines  are  F,  —  X,  Z  and  which  passes 
through  the  point  (x,  y,  0)  generates  a  normal  congruence. 

2.  Prove  that  the  tangent  planes  to  two  confocal  quadrics  at  the  points  of  con 
tact  of  a  common  tangent  are  perpendicular,  and  consequently  that  the  common 
tangents  to  two  confocal  quadrics  form  a  normal  congruence. 

3.  Find  the  congruence  of  common  tangents  to  the  paraboloids 

x'2  +  y'2  =  2az,         x2  +  y*  =  -  2  az, 
and  determine  the  focal  surfaces. 

4.  If  two  ruled  surfaces  through  a  line  L  are  represented  on  the  sphere  by 
orthogonal  lines,  their  lines  of  striction  meet  L  at  points  equally  distant  from  the 
middle  point. 

5.  In  order  that  the  focal  planes  for  each  line  of  a  congruence  meet  under  the 
same  angle,  it  is  necessary  and  sufficient  that  the  osculating  planes  of  the  edges  of 
regression  of  the  developables  meet  the  tangent  planes  to  the  focal  surfaces  under 
constant  angle. 

6.  A  necessary  and  sufficient  condition  that  a  surface  of  reference  of  a  congru 
ence  be  its  middle  surface  is  g£  -  (/  +  /  )<^+  e&  '=  0. 

164.  Associate  normal  congruences.    If  we  put 

dx  dx 


equations  (8)  may  be  replaced  by 

(34)  t  =  c  —  I  7  du  +  y^dv, 

where  c  is  a  constant.    Now  equation  (9)  is  equivalent  to 


402  RECTILINEAR  CONGRUENCES 

In  consequence  of  this  condition  equation  (34)  may  be  written 

(36)  t  =  c-$(u^ 

where  ul  is  a  function  of  u  and  v  thus  denned.  If  the  orthogonal 
trajectories  of  the  curves  u^  const,  be  taken  as  parametric  curves 
vl=  const.,  it  follows  from  (36)  and  from  equations  in  ul  and  vl 
analogous  to  (33)  and  (34)  that 


From  this  result  follows  the  theorem  : 

The  lines  of  a  normal  congruence  cut  orthogonally  the  curves  on 
the  surface  of  reference  at  whose  points  t  is  constant. 

If  0  denotes  the  angle  which  a  line  of  the  congruence  makes 
with  the  normal  to  the  surface  of  reference  at  the  point  of  inter 
section,  we  have 

(37)  sin*= 


where  the  linear  element  of  the  surface  is 


If  S  be  taken  for  the  surface  of  reference  of  a  second  congruence 
whose  direction-cosines  Xv  Yv  Zl  satisfy  the  conditions 


where  4>i(ui)  ig  anv  function  whatever  of  u^  this  congruence  is 
normal  and       has  the  value 


Since  01  is  any  function,  there  is  a  family  of  these  normal  congru 
ences  which  we  call  the  associates  of  the  given  congruence  and  of 
one  another.  Through  any  point  of  the  surface  of  reference  there 
passes  a  line  of  each  congruence,  and  all  of  these  lines  lie  in  the 
plane  normal  to  the  curve  ul  —  const,  through  the  point.  Hence  : 

The  two  lines  of  two  associate  congruences  through  the  same  point 
of  the  surface  of  reference  lie  in  a  plane  normal  to  the  surface. 


DERIVED  CONGRUENCES  403 

Combining  with  equation  (37)  a  similar  one  for  an  associate  con 
gruence,  we  have 

(38)  «E*=£&)=/(W) 

sin^     #K)     '<*>'• 

Hence  we  have  the  theorem : 

The  ratio  of  the  sines  of  the  angles  which  the  lines  of  two  associate 
congruences  make  with  the  normal  to  their  surface  of  reference  is  con 
stant  along  the  curves  at  whose  points  t  is  constant. 

When  in  particular  f(u^  in  (38)  is  a  constant,  the  former  theorem 
and  equation  (38)  constitute  the  laws  of  reflection  and  refraction  of 
rays  of  light,  according  as  the  constant  is  equal  to  or  different  from 
minus  one.  And  so  we  have  the  theorem  of  Malus  and  Dupin : 

If  a  bundle  of  rays  of  light  forming  a  normal  congruence  be  reflected 
or  refracted  any  number  of  times  by  the  surfaces  of  successive  homo 
geneous  media,  the  rays  continue  to  constitute  a  normal  congruence. 

By  means  of  (37)  equation  (36)  can  be  put  in  the  form 
t  =  c—l  \l  E  sin  6  dur 

From  this  result  follows  the  theorem  of  Beltrami  *  : 

If  a  surface  of  reference  of  a  normal  congruence  be  deformed  in  such 
a  way  that  the  directions  of  the  lines  of  the  congruence  with  respect 
to  the  surface  be  unaltered,  the  congruence  continues  to  be  normal. 

165.  Derived  congruences.  It  is  evident  that  the  tangents  to  the 
curves  of  any  one-parameter  family  upon  a  surface  S  constitute  a 
congruence.  If  these  curves  be  taken  for  the  parametric  lines 
v  =  const.,  and  their  conjugates  for  u  =  const.,  the  developables  in 
one  family  have  the  curves  v  =  const,  for  edges  of  regression,  and 
the  developables  of  the  other  family  envelop  S  along  the  curves 
u  —  const.  We  may  take  S  for  the  surface  of  reference.  If  Sl  be 
the  other  focal  surface,  the  lines  of  the  congruence  are  tangent  to 
the  curves  u  =  const,  on  Sr  The  tangents  to  the  curves  v  =  const, 
on  S1  form  a  second  congruence  of  which  Sl  is  one  focal  surface, 
and  the  second  surface  $2  is  uniquely  determined.  Moreover,  the 

*  Giornale  di  matematiche,  Vol.  II  (1864),  p.  281. 


404  RECTILINEAR  CONGRUENCES 

lines  of  the  second  congruence  are  tangent  to  the  curves  u  =  const. 
on  Sz.  In  turn  we  may  construct  a  third  congruence  of  tangents 
to  the  curves  v  —  const,  on  Sz.  This  process  may  be  continued 
indefinitely  unless  one  of  these  focal  surfaces  reduces  to  a  curve, 
or  is  infinitely  distant. 

In  like  manner  we  get  a  congruence  by  drawing  tangents  to 
the  curves  u  =  const,  on  S,  which  is  one  focal  surface,  and  the 
other,  S_v  is  completely  determined.  The  tangents  to  the  curves 
u  —  const,  on  S_l  form  still  another,  and  so  on.  In  this  way  we 
obtain  a  suite  of  surfaces 


which  is  terminated  only  when  a  surface  reduces  to  a  curve,  or 
its  points  are  infinitely  distant.  Upon  each  of  these  surfaces  the 
parametric  curves  form  a  conjugate  system.  The  congruences  thus 
obtained  have  been  called  derived  congruences  by  Darboux.*  It  is 
clear  that  the  problem  of  finding  all  the  derived  congruences  of  a 
given  one  reduces  to  the  integration  of  the  equation  of  its  devel- 
opables  (25);  for,  when  the  developables  are  known  we  have  the 
conjugate  system  on  its  focal  surfaces. 

In  order  to  derive  the  analytical  expressions  for  these  results, 
we  recall  (§  80)  that  the  coordinates  x,  y,  z  of  S  are  solutions 
of  an  equation  of  the  form 

(39) 


du  dv         du         cv 

where  a  and  b  are  determinate  functions  of  u  and  v.  If  the  coordi 
nates  of  Sl  be  denoted  by  x^  y^  2t,  they  are  given  by 

dx  By  ,  ,    dz 

*-x+\-*       fc-jr  +  x,-.       v'  +  XiS. 

where  \^J~E  measures  the  distance  between  the  focal  points.  But 
as  the  lines  of  the  congruence  are  tangent  to  the  curves  u  =  const. 
on  Sv  we  must  have 

dx.          dx  dy.  dy  dz.  dz 

(40)  —  1  =  M1  —  »         -^I  =  u1—  *         —  i  =  u  —  , 

dv     Pldu         dv     Pldu         dv      ldu 

*  Vol.  II,  pp.  16-22. 


DERIVED  CONGRUENCES  405 

where  /-^  is  a  determinate  function  of  u  and  v.  When  the  above 
value  for  xl  is  substituted  in  the  first  of  these  equations,  the  result 
is  reducible,  by  means  of  (39),  to 

L^LI  _  flX      n\te  +  (1  __  fog  to  =  0. 

dv  V  du  l/  dv 

Since  the  same  equation  is  true  for  y  and  z,  the  quantities  in  paren 
theses  must  be  zero,  that  is, 

1  a  1      a 


Hence  the  surface  Sl  is  defined  by 

,  1  ex  \dy  ,  1  dz 


and  equations  (40)  become 


/42\       i  •__  __ 

^y      Vav  b      b]du     dv      \dv  b      b/du'    dv      \dv  b      bjdu 

Proceeding  in  a  similar  manner,  we  find  that  $_i  is  defined  by 
the  equations 

/4ox  -  1  a*  -  1  to  .  .  1  * 

V*°;       »- 

and  that 


and  similar  expressions  in  y_^  and  2_i. 

From  (41)  and  (43)  it  is  seen  that  the  surface  Sl  or  S.i  is  at 
infinity,  according  as  b  or  a  is  zero.  When  a  and  5  are  both  zero, 
S  is  a  surface  of  translation  (§  81).  Hence  the  tangents  to  the 
generators  of  a  surface  of  translation  form  two  congruences  for 
each  of  which  the  other  focal  surface  is  at  infinity. 

In  order  that  S^  be  a  curve,  x^  y^  zl  must  be  functions  of  u  alone. 
From  (42)  it  follows  that  the  condition  for  this  is 

d  1  _a 

~dv  5  ~6* 

In  like  manner  the  condition  that  $_i  be  a  curve  is 

±l=i 

du  a      a 


406  RECTILINEAR  CONGRUENCES 

The  functions  h  and  &,  denned  by 

h__da        ,  if  —  ®* 

du  dv 

are  called  the  invariants  of  the  differential  equation  (39).    Hence 
the  above  results  may  be  stated : 

A  necessary  and  sufficient  condition  that  the  focal  surface  Sl  or 
£_i  be  a  curve  is  that  the  invariant  k  or  h  respectively  of  the  point 
equation  of  S  be  zero. 

166.  Fundamental  equations  of  condition.  We  have  seen  (§  160) 
that  with  every  congruence  there  are  associated  two  quadratic  dif 
ferential  forms.  Now  we  shall  investigate  under  what  conditions 
two  quadratic  forms  determine  a  congruence.  We  assume  that  we 
have  two  such  forms  and  that  there  is  a  corresponding  congruence. 
The  tangents  to  the  parametric  curves  on  the  surface  of  reference 
at  a  point  are  determined  by  the  angles  which  they  make  with  the 
tangents  to  the  parametric  curves  of  the  spherical  representation  of 
the  congruence  at  the  corresponding  point,  and  with  the  normal  to 
the  unit  sphere.  Hence  we  have  the  relations 

,__._+„«• 

I  Zti  7)ti 

(44) 


and  similar  equations  in  y  and  2,  where  #,  /3,  7;  ar  (Sv  ryl  are  functions 

of  u  and  v.    If  we  multiply  these  equations  by  — >  — »  —  respec- 

dX   dY    dZ  du     dU    3U 

tively,  and  add;  also  by  — *  — »  —  and  by  A",  Y,  Z\  we  obtain 

dv     dv     dv 


from  which  we  derive 

e&     j  c/  j  iy      c-c/ 

a  =         •    ^2*  p=- — *     7  =^.A — > 

/  \  f  \  (O&  ~~  cy  (n  &  ~~~  c/  01^ 

(45) 


Ctf  «M*        fl  CV^  *>2'  '1 

foo/  —  CA  (Q^/  —  c/1 


yi^^rr 


FUNDAMENTAL  EQUATIONS  407 

In  order  that  equations  (44)  be  consistent,  we  must  have 


du  \dv        2v  \du 
which,  in  consequence  of  equations  (V,  22),  is  reducible  to  the  form 

]t?X+Sd-X 

du          dv 

where  J?,  S,  T  are  determinate  functions.  Since  this  equation  must 
be  satisfied  by  Y  and  Z  also,  we  must  have  R  —  0,  S=  0,  T=  0. 
When  the  values  of  a,  y3,  a^  fiv  from  (45),  are  substituted  in  these 
equations,  we  have 


(47) 


Conversely,  when  we  have  a  quadratic  form  whose  curvature 
is  +1,  it  may  be  taken  as  the  linear  element  of  the  spherical  rep 
resentation  of  a  congruence,  which  is  determined  by  any  set  of 
functions  e,  f,  /',  #,  7,  7^  satisfying  equations  (4T).  For,  when 
these  equations  are  satisfied,  so  also  is  (46),  and  consequently 
the  coordinates  of  the  surface  of  reference  are  given  by  the 
quadratures  (44). 

Incidentally  we  remark  that  when  the  congruence  is  normal,  and 
the  surface  of  reference  is  one  of  the  orthogonal  surfaces,  the  last 
of  equations  (47)  is  satisfied  identically,  and  the  first  two  reduce 
to  the  Codazzi  equations  (V,  27). 

We  apply  these  results  to  the  determination  of  the  congruences 
with  an  assigned  spherical  representation  of  their  principal  surfaces, 
and  those  with  a  given  representation  of  their  developables. 

167.  Spherical  representation  of  principal  surfaces  and  of  devel 
opables.  A  necessary  and  sufficient  condition  that  the  principal 
surfaces  of  a  congruence  cut  the  surface  of  reference  in  the  para 
metric  lines  is  given  by  (21). 


408  RECTILINEAR  CONGRUENCES 

If  we  require  that  the  surface  of  reference  be  the  middle  surface 
of  the  congruence,  and  if  r  denote  half  the  distance  between  the 
limit  points,  we  have,  from  (15), 

(48)  e  —  —  r<o,         g  =  r& 

When  these  values  are  substituted  in  (47),  the  first  two  become 

12,          f?  a/  /  \ 


(49) 


1    d  , 
7  =  ^(r< 


and  the  last  is  reducible  to  * 

,50)  2  av   ial°g^g?-  i  aiog^gTjg'iog^,. 

dw£y          dv      cu          du      dv         dudv 

d\    \$   d  i    f    YL  a  F    W  d  {    f    l\4-2f      0 

+sLNtfsfe;J+4N?s\^Jr^  * 

Moreover,  equations  (44)  become 

ao:         ax    /  ax  ,  aa:    /  ax      ax 


where  7  and  yl  are  given  by  (49)  ;  and  similar  equations  in  y  and  z. 
Our  problem  reduces,  therefore,  to  the  determination  of  pairs  of 
functions  r  and  /  which  satisfy  (50).  Evidently  either  of  these 
functions  may  be  chosen  arbitrarily  and  the  other  is  found  by  the 
solution  of  a  partial  differential  equation  of  the  second  order. 
Hence  any  orthogonal  system  on  the  unit  sphere  serves  for  the 
representation  of  the  principal  surfaces  of  a  family  of  congruences, 
whose  equations  involve  three  arbitrary  functions. 

In  order  that  the  parametric  curves  on  the  sphere  represent  the 
developables  of  a  congruence,  it  is  necessary  and  sufficient  that 


as  is  seen  from  (25).  If  the  surface  of  reference  be  the  middle  sur 
face,  and  p  denotes  half  the  distance  between  the  focal  points,  it 
follows  from  (15)  that  e 


p       "  c      $ 


*  Cf.  Bianchi,  Vol.  I,  p.  314. 


DEVELOPABLES  PARAMETRIC  409 

Combining  these  equations  with  the  above,  we  have 
(52)  e=-p&        f  =  -f'  =  p&         ff  =  p& 

When  these  values  are  substituted  in  the  first  two  of  equations  (47) 
and  the  resulting  equations  are  solved  for  7  and  7^  we  find 


and  the  last  of  equations  (47)  reduces  to 

d  ri2V    a  n2 


= 

Each  solution  of  this  equation  determines  a  congruence  with  the 
given  representation  of  its  developables,*  and  the  middle  surface 
is  given  by  the  quadratures 


(54) 


and  similar  expressions  in  y  and  2. 

When  the  values  (52)  are  substituted  in  (18)  the  latter  becomes 


Consequently  equation  (32)  reduces  to 

a      2P 

sin  ^  =  -^  = 


Referring  to  equation  (III,  16),  we  have: 

The  angle  between  the  focal  planes  of  a  congruence  is  equal  to  the 
angle  between  the  lines  on  the  sphere  representing  the  corresponding 
developables. 

This  result  is  obtained  readily  from  geometrical  considerations. 

168.  Fundamental  quantities  for  the  focal  surfaces.  We  shall 
make  use  of  these  results  in  deriving  the  expressions  for  the  funda 
mental  quantities  of  the  focal  surfaces  Sl  and  $2,  which  are  defined  by 


*  This  result  is  due  to  Guichard,  Annales  de  I'Ecole  Normale,  Ser.  3,  Vol.  VI  (1889), 
pp.  342-344. 


410  RECTILINEAR  CONGRUENCES 

From  these  and  (54)  we  get 


The  coefficients  of  the  linear  elements  of  ^  and  £2,  as  derived 
from  these  formulas,  are 


(56) 

and 

(57)  . 


The  direction-cosines  of  the  normals  to  ^  and  S2  denoted  by 
X^  Yv  Z^  JT2,  r2,  ^2  respectively  are  found  from  the  above  equa 
tions  and  (V,  31)  to  have  the  values 


Si)  =._        3V  ^V 

\  d(u,  v) 


•v  .  ^  _ 

1  ZJ-       2/^,      «i\  /~^  ^/^s'/./V^      /}?>  '      ^ 


and  similar  expressions  for  Yt  and  Z{.  If  these  equations  be  differ 
entiated,  and  the  resulting  equations  be  reduced  by  means  of  (V,  22), 
they  can  be  put  in  the  form 


K  \\  fi2Vax       I     a*t_^/22Vwr 
"VL^^lJ"^    x\     dv-^lif  dv 


^^ 

"a^"==Til2/ 


FOCAL  SURFACES 
From  these  expressions  and  (55)  we  obtain 


(58) 


and 


411 


^  du  du  ~       V^  \du  '   ^2 

D[  =  — Y  ^£1  Mi  =  _  y  ?£i  Mi  =  o, 

^  dv   du          ^  du    cv 

A"= ~2  ^r  "^7  =  * 


From  the  foregoing  formulas  we  derive  the  following  expressions 
for  the  total  curvature  of  Sl  and  of  Sz: 


(60) 


{ 


22V 
1  J 


EXAMPLES 

1.  If  upon  a  surface  of  reference  S  of  a  normal  congruence  the  curves  orthog 
onal  to  the  lines  of  the  congruence  are  defined  by  0(u,  u)  =  const.,  and  6  denotes 
the  angle  between  a  line  of  the  congruence  and  the  normal  to  the  surface  at  the 
point  of  meeting,  then  sin2  0  =  AiF(0)  where  the  differential  parameter  is  formed 
with  respect  to  the  linear  element  of  S.  Show  that  6  is  constant  along  a  line  0  =  const. 
only  when  the  latter  is  a  geodesic  parallel. 

2.  When  in  the  point  equation  of  a  surface,  namely 

c20    ,      c0  ,  ,  30      n 
-  +  a—  +  6—  =  0, 
du  cv         du        cv 

a  or  6  is  zero,  the  coordinates  of  the  surface  can  be  found  by  quadratures. 

3.  Find  the  derived  congruences  of  the  tangents  to  the  parametric  curves  on  a 
tetrahedral  surface  (Ex.  2,  p.  267),  and  determine  under  what  conditions  the  sur 
face  Si  or  5-i  is  a  curve. 

4.  Find  the  equation  of  the  type  (39)  which  admits  as  solutions  the  quantities 
*i,  yi,  zi  given  by  (41). 

5.  When  a  congruence  consists  of  the  tangents  to  the  lines  of  curvature  in 
one  system  on  a  surface,  the  focal  distances  are  equal  to  the  radii  of  geodesic 
curvature  of  the  lines  of  curvature  in  the  other  system. 


412  RECTILINEAR  CONGRUENCES 

6.  Let  S  be  a  surface  referred  to  its  lines  of  curvature,  let  «i  and  s2  denote  the 
arcs  of  the  curves  v  =  const,  and  u  =  const,  respectively,  ri  and  r2  their  radii  of 
first  curvature,  and  RI  and  JR2  their  radii  of  geodesic  curvature  ;  for  the  second 
focal  sheet  Si  of  the  congruence  of  tangents  to  the  curves  v  =  const,  the  linear 
element  is  reducible  to  2 


hence  the  curves  Si  =  const,  are  geodesies. 

7.  Show  that  2t  of  Ex.  6  is  developable  when  n  =/(si),  and  determine  the 
most  general  form  of  r\  so  that  2i  shall  be  developable. 

8.  Determine  the  condition  which  p  must  satisfy  in  order  that  the  asymptotic 
lines  on  either  focal  surface  of  a  congruence  shall  correspond  to  a  conjugate  system 
on  the  other,  and  show  that  in  this  case 


where  0  denotes  the  angle  between  the  focal  planes. 

9.  In  order  that  the  focal  surfaces  degenerate  into  curves,  it  is  necessary  and 
sufficient  that  the  spherical  representation  satisfy  the  conditions 

—  {12\'=—{12\'=  (12 
du  \  1  )       cv  \  2  }  ~  \  1 

10.  Show  that  the  surfaces  orthogonal  to  a  normal  congruence  of  the  type  of 
Ex.  9  are  cyclides  of  Dupin. 

11.  A  necessary  and  sufficient  condition  that  the  second  sheet  of  the  congruence 
of  tangents  to  a  family  of  curves  on  a  surface  S  be  developable  is  that  the  curves 
be  plane. 

169.  Isotropic  congruences.  An  isotropic  congruence  is  one  whose 
focal  surfaces  are  developables  with  minimal  edges  of  regression. 
In  §  31  we  saw  that  H  =  0  is  a  necessary  and  sufficient  condition 
that  a  surface  be  of  this  kind.  Referring  to  (56)  and  (57),  we  see 
that  we  must  have 


From  (54)  it  is  seen  that  if  p  were  zero  the  middle  surface  would 
be  a  point,  and  from  (55)  that  if  the  expressions  in  parentheses 
were  zero  the  surfaces  Sl  and  $2  would  be  curves.  Consequently 

(61)  <£•  =  g  =  0. 

Conversely,  if  this  condition  be  satisfied,  Sl  and  S.2  are  isotropic 
developables.  Hence  an  isotropic  congruence  is  one  whose  devel 
opables  are  represented  on  the  sphere  by  minimal  lines. 


ISOTROPIC  CONGRUENCES  413 

In  consequence  of  (61)  we  have,  from  (52), 

and  since  f+f  also  is  zero,  it  follows  that 

(62)  dxdX+  dydY+  dzdZ=  0. 

Therefore  r  is  zero,  so  that  all  the  lines  of  striction  lie  on  the 
middle  surface.  Since  (61)  is  a  consequence  of  (62),  we  have 
the  following  theorem  of  Ribaucour,*  which  is  sometimes  taken 
for  the  definition  of  isotropic  congruences : 

All  the  lines  of  striction  of  an  isotropic  congruence  lie  on  the  mid 
dle  surface  ;  and,  conversely,  when  all  the  lines  of  striction  lie  on  the 
middle  surface,  the  congruence  is  isotropic  ;  moreover,  the  middle  sur 
face  corresponds  to  the  spherical  representation  with  orthogonality  of 
linear  elements. 

Ribaucour  has  established  also  the  following  theorem :  f 

TJie  middle  envelope  of  an  isotropic  congruence  is  a  minimal  surface. 

Since  the  minimal  lines  on  the  sphere  are  parametric,  in  order 
to  prove  this  theorem  it  is  only  necessary  to  show  that  on  the 
middle  envelope,  that  is,  the  envelope  of  the  middle  planes, 
the  corresponding  lines  form  a  conjugate  system.  If  W  denotes 
the  distance  of  the  middle  plane  from  the  origin,  the  condition 
necessary  and  sufficient  that  the  parametric  lines  be  conjugate 
is  that  W  satisfy  the  equation 

(63)  r  +  <^0  =  0. 
By  definition 

and  with  the  aid  of  (V,  22)  we  find 

ft 


du  dv      cu  dv 

o2 

Since   equation   (53)   reduces   to  — £-  +  /><^=0,  the  function  W> 
satisfies  (63). 

*  Etude  des  Elasso'ides  ou  Surfaces  a  Courbure  Moyenne  Nulle,  Memoires  Couronnts 
par  r  Academic  de  Belgique,  Vol.  XLIV  (1881),  p.  63.  t  L.c.,  p.  31. 


414  RECTILINEAR  CONGRUENCES 

170.  Congruences  of  Guichard.  Guichard*  proposed  and  solved 
the  problem  : 

To  determine  the  congruences  whose  focal  surfaces  are  met  by  the 
developables  in  the  lines  of  curvature. 

With  Bianchi  we  call  them  congruences  of  Gruichard. 

We  remark  that  a  necessary  and  sufficient  condition  that  a  con 
gruence  be  of  this  kind  is  that  Fl  and  F2  of  §  168  be  zero.  From 
(56)  and  (57)  it  is  seen  that  this  is  equivalent  to 

Comparing  this  result  with  §  78,  we  have  the  theorem: 

A  necessary  and  sufficient  condition  that  the  developables  of  a  con 
gruence  meet  the  focal  surfaces  in  their  lines  of  curvature  is  that  the 
congruence  be  represented  on  the  sphere  by  curves  representing  also 
the  asymptotic  lines  on  a  pseudospherical  surface. 

In  this  case  the  parameters  can  be  so  chosen  thatf 

<£F=^=1,  c?  =  —  COSQ), 

where  co  is  a  solution  of 


=  sin  ft). 


dudv 


In  this  case  equation  (53)  is 

(65)  —  £-  =  p  cos  <0. 


In  particular,  this  equation  is  satisfied  by  X,  F,  Z  (V,  22).    If  we 
replace  p  by  X  in  (54),  we  have 


consequently,  for  the  congruence  determined  by  this  value  of  />, 
the  middle  surface  is  a  plane. 

From  (55)  it  follows  that  the  lines  of  the  congruence  are  tangent 
to  the  lines  of  curvature  v  =  const,  on  *Sy    Consequently  they  are 

*L.c.,  p.  346.  f  This  is  the  only  real  solution  of  (64). 


CONGRUENCES  OF  GUICHARD  415 

parallel  to  the  normals  to  one  of  the  sheets  of  the  evolute  of  Sl 
(cf.  §  74) ;  call  it  2X.  Hence  the  conjugate  system  on  2t  corre 
sponding  to  the  lines  of  curvature  on  S^  is  represented  on  the 
sphere  by  the  same  lines  as  the  developables  of  the  congruence. 
Referring  to  (VI,  38),  we  see  that  condition  (64)  is  equivalent  to 


where  the  Christoffel  symbols  are  formed  with  respect  to  the  linear 
element  of  2r  But  these  are  the  conditions  that  the  parametric 
curves  on  2X  be  geodesies  (cf.  §  85).  Surfaces  with  a  conjugate 
system  of  geodesies  were  studied  by  Voss,  *  and  on  this  account 
are  called  surfaces  of  Voss.  Since  the  converse  of  the  above  results 
is  true,  we  have  the  following  theorem  of  Guichard : 

A  necessary  and  sufficient  condition  that  the  tangents  to  the  lines 
of  curvature  in  one  family  of  a  surface  form  a  congruence  of 
Guichard  is  that  one  sheet  of  the  evolute  of  the  surface  be  a  sur 
face  of  Voss,  and  that  the  tangents  constituting  the  congruence  be 
those  which  are  parallel  to  the  normals  to  the  latter. 

If  Wl  denotes  the  distance  from  the  origin  to  the  tangent  plane 
to  the  surface  of  Voss  2X,  then  Wl  is  a  solution  of  equation  (65) 
(cf.  §  84).  Hence  W^  +  Kp  is  a  solution  of  this  equation,  provided  K 
be  a  constant.  But  since  the  tangent  plane  to  2X  passes  through 
the  corresponding  point  of  Sv  the  above  result  shows  that  a  plane 
normal  to  the  lines  of  the  congruence,  and  which  divides  in  con 
stant  ratio  the  segment  between  the  focal  points,  envelopes  a  sur 
face  of  Voss.  In  particular,  we  have  the  corollary : 

The  middle  envelope  of  a  congruence  ofGruichardis  a  surface  of  Voss. 

171.  Pseudospherical  congruences.  The  lines  joining  correspond 
ing  points  on  a  pseudospherical  surface  S  and  on  one  of  its  Backhand 
transforms  S1  (cf.  §  120)  constitute  an  interesting  congruence.  We 
recall  that  the  distance  between  corresponding  points  is  constant, 
and  that  the  tangent  planes  to  the  two  surfaces  at  these  points 
meet  under  constant  angle.  From  (32)  it  follows  that  the  distance 
between  the  limit  points  also  is  constant. 

*Miinchener  JSerichte,  Vol.  XVIII  (1888),  pp.  95-102. 


416  KECTILINEAB  CONGRUENCES 

Conversely,  when  the  angle  between  the  focal  planes  of  a  con 
gruence  is  constant,  and  consequently  also  the  angle  6  between 
the  parametric  lines  on  the  sphere  representing  the  developables, 
we  have,  from  (V,  4), 

111112  i 


Furthermore,  if  the  distance  between  the  focal  points  is  constant,  we 
have  p  =  a,  and  by  (60)  gjn2  Q 

K^  =  K*=    "4^" 

Hence  the  two  focal  surfaces  have  the  same  constant  curvature. 
Congruences  of  this  kind  were  first  studied  by  Bianchi.*  He 
called  them  pseudospherical  congruences. 

In  order  that  the  two  focal  surfaces  of  the  congruence  be  Back- 
lund  transforms  of  one  another,  it  is  necessary  that  their  lines  of 
curvature  correspond.  It  is  readily  found  that  for  both  surfaces 
the  equation  of  these  lines  is  reducible  by  means  of  (66)  to 


{  12V  f  12V  H2V2       n^v2! 

is\2jdw'~[^+  f\is  +  n  2  )  J 


Moreover,  the  differential  equation  of  the  asymptotic  lines  on  each 
surface  is  £dv?—  ^/di)2—  0.  Hence  we  have  the  theorems: 

On  the  focal  surfaces  of  a  pseudospherical  congruence  the  lines  of 
curvature  correspond,  and  likewise  the  asymptotic  lines. 

The  focal  surfaces  of  a  pseudospherical  congruence  are  Backlund 
transforms  of  one  another. 

EXAMPLES 

1.  When  the  parameters  of  a  congruence  are  any  whatsoever,  and  likewise  the 
surface  of  reference,  a  condition  necessary  and  sufficient  that  a  congruence  be 
isotropic  is  e        f  +  f       g 

~€~    2^    =^' 

2.  A  necessary  and  sufficient  condition  that  a  congruence  be  isotropic  is  that 
the  locus  of  two  points  on  each  line  at  an  equal  constant  distance  from  the  middle 
surface  shall  describe  applicable  surfaces. 

3.  Show  that  equation  (65)  admits  —  and  —  as  solutions.   Prove  that  in  each 

3u          dv 
case  one  of  the  focal  surfaces  is  a  sphere. 

*Annali,  Ser.  2,  Vol.  XV  (1887),  pp.  161-172;  also  Lezioni,  Vol.  I,  pp.  323,  324. 


JF-CONGRUENCES 


41T 


4.  Determine  all  the  congruences  of  Guichard  for  which  one  of  the  focal  surfaces 
is  a  sphere. 

5.  When  a  surface  is  referred  to  its  lines  of  curvature,  a  necessary  and  suffi 
cient  condition  that  the  tangents  to  the  curves  v  =  const,  shall  form  a  congruence 
of  Guichard  is 


a  /  1 
3u\^ 


6.  Determine  the  surfaces  which  are  such  that  the  tangents  to  the  lines  of 
curvature  in  each  system  form  a  congruence  of  Guichard. 

172.  TF-congruences.  We  have  just  seen  that  the  asymptotic  lines 
on  the  focal  surfaces  of  a  pseudospherical  congruence  correspond  ; 
the  same  is  true  in  the  case  of  the  congruences  of  normals  to  a 
JF-surface  (cf.  §  124).  For  this  reason  all  congruences  possessing 
this  property  are  called  W-congruences.  We  shall  derive  other  prop 
erties  of  these  congruences. 

The  condition  that  asymptotic  lines  correspond,  namely 


takes  the  following  form  in  consequence  of  (58)  and  (59): 

22V 

Hence  from  (60)  it  follows  that  a  necessary  and  sufficient  condition 
for  a  JF-congruence  is 


In  order  to  obtain  an  idea  of  the  analytical  problem  involved  in 
the  determination  of  TF-congruences,  we  suppose  that  we  have  two 
surfaces  £,  S  referred  to  their  asymptotic  lines,  and  inquire  under 
what  conditions  the  lines  joining  corresponding  points  on  the  surfaces 
are  tangent  to  them.  We  assume  that  the  coordinates  of  the  surfaces 
are  defined*  by  means  of  the  Lelieuvre  formulas  (cf.  §  79),  thus: 


(68) 


dx 

du 

dx_ 

du 


du     du 


du    du 


dx 


dx 


dv     dv 


~dv     dv 


*Cf.  Guichard,  Comptes  Rendus,  Vol.  CX  (1890),  pp.  126-127. 


418  RECTILINEAR  CONGRUENCES 

and  similar  equations  in  y,  z,  y,  and  z.    The  functions  vv 
vv  i>2,  vs  respectively  are  solutions  of  equations  of  the  form 


(69) 


dudv 
and  they  are  such  that 

(70)  v?  +  »l  +  vl  =  a,          vl  +  v$  +  vl  =  a, 
wliere  a  and  a  are  defined  by 

(71)  J5T  =  -—  ,,         K  =  -~. 

a-  a2 

Since  v^  v^  vs  and  vr  vz,  vs  are  proportional  to  the  direction- 
cosines  of  the  normals  to  S  and  S,  the  condition  that  the  lines 
joining  corresponding  points  be  tangent  to  the  surfaces  S  and 
S  is 


v^x  -z)+  vz(y  -y}+  v^(z  -  z)  =  0. 
Hence 


x  —  x  y  —  v  z  —  z 


where  w  denotes  a  factor  of  proportionality.     In  order  to  find  its 
value,  we  notice  that  from  these  equations  follow  the  relations 

(2  />)2=  ^(x  -  x)*=  7?i22(^3-  iy>2)a 


=  w 


where  ^  denotes  the  angle  between  the  focal  planes.  If  this  value 
of  2p  and  the  values  of  K  and  K  from  (71)  be  substituted  in  (67), 
it  is  found  that  m*  =  1.  We  take  w=l,  thus  fixing  the  signs  of 
i/j,  i>2,  i>3,  and  the  above  equations  become 

(72)      x  —  x  =  V&—  vfa     y  —  y  =  v^t—  *&,    z  -  *  =  W-  v2vr 

If  the  first  of  these  equations  be  differentiated  with  respect  to  w, 
the  result  is  reducible  by  (68)  to 


JF-CONGRUENCES 


419 


Proceeding  in  like  manner  with  the  others,  and  also  differentiating 
with  respect  to  v,  we  are  brought  to 

0       /-  7      /- 


(73) 


.  =  1,2,8) 


where  Z  and  &  are  factors  of  proportionality  to  be  determined. 

If  the  first  of  these  equations  be  differentiated  with  respect  to  v, 
and  in  the  reduction  we  make  use  of  the  second  and  of  (69),  we  find 


In  like  manner,  if  the  second  of  the  above  equations  be  differen 
tiated  with  respect  to  u,  we  obtain 


Since  these  equations  are  true  for  i=l,  2,  3,  the  quantities  in 
parentheses  must  be  zero. ,  This  gives 

cl      3k 
du      dv 


x=-^+ 


In  accordance  with  the  last  we  put 


and  the  others  become 


a ,    i 

=  —  IQOT— - 
* 


Hence  equations  (69)  may  be  written 


Bdudv 


dudv        ldufo\0. 


i  \  if 

from  which  it  follows  that  0l  is  a  solution  of  the  first  of  equa 
tions  (69)  and  l/0l  of  the  second.  Moreover,  equations  (73)  may 
now  be  written  in  the  form 


0,      v< 


du 


0 


dv 


420  RECTILINEAR  CONGRUENCES 

Hence  if  Ql  be  a  known  solution  of  the  first  of  equations  (69),  we 
obtain  by  quadratures  three  functions  vf,  which  lead  by  the  quadra 
tures  (68)  to  a  surface  S.  The  latter  is  referred  to  its  asymptotic 
lines  and  the  joins  of  corresponding  points  on  S  and  $  are  tangent 
to  the  latter.  And  so  we  have : 

If  a  surface  S  be  referred  to  its  asymptotic  lines,  and  the  equations 
of  the  surface  be  in  the  Lelieuvre  form,  each  solution  of  the  corre 
sponding  equation  ffQ 

=  \0 

dudv 

determines  a  surface  S,  found  by  quadratures,  such  that  S  and  S  are 
the  focal  surfaces  of  a  W-congruence. 

Comparing  (74)  with  (XI,  13),  we  see  that  if  we  put 

^1=^1.         yi=0i»»         «i=^8» 

the  locus  of  the  point  (x^  y^  zj  corresponds  to  S  with  orthogo 
nality  of  linear  elements.  Hence  vv  v2,  vs  are  proportional  to  the 
direction-cosines  of  the  generatrices  of  an  infinitesimal  deformation 
of  £,  so  that  we  have : 

Each  focal  surface  of  a  W-congruence  admits  of  an  infinitesimal 
deformation  whose  generatrices  are  parallel  to  the  normals  to  the 
other  focal  surface. 

Since  the  steps  in  the  preceding  argument  are  reversible,  we 
have  the  theorem : 

The  tangents  to  a  surface  which  are  perpendicular  to  the  genera 
trices  of  an  infinitesimal  deformation  of  the  latter  constitute  a  W- 
congruence  of  the  most  general  kind ;  and  the  normals  to  the  other 
surface  are  parallel  to  the  generatrices  of  the  deformation. 

173.  Congruences  of  Ribaucour.  In  his  study  of  surfaces  corre 
sponding  with  orthogonality  of  linear  elements  Ribaucour  consid 
ered  the  congruence  formed  by  the  lines  through  points  on  one 
surface  parallel  to  the  normals  to  a  surface  corresponding  with  the 
former  in  this  manner.  Bianchi  *  calls  such  a  congruence  a  con 
gruence  of  Ribaucour,  and  the  second  surface  the  director  surface. 

In  order  to  ascertain  the  properties  of  such  a  congruence,  we 
recall  the  results  of  §  153.  Let  Sl  be  taken  for  the  surface  of 

*Vol.  II,  p.  17. 


CONGRUENCES  OF  RIBAUCOUR  421 

reference,  and  draw  lines  parallel  to  the  normals  to  S.    If  the  latter 
be  referred  to  its  asymptotic  lines,  it  follows  from  (XI,  6)  that 


du  dv 
Since  these  values  satisfy  the  conditions 


civ  du        HK 

=y  dxldX 

9  ~^ 


the  ruled  surfaces  u  =  const.,  v  =  const,  are  the  developables.  And 
since  also  p^+  p2  is  equal  to  zero,  Sl  is  the  middle  surface  of  the 
congruence.  But  the  parametric  curves  on  8t  form  a  conjugate 
system  when  the  asymptotic  lines  on  S  are  parametric.  Hence  we 
have  the  theorem  : 

The  developable  surfaces  of  a  congruence  of  Ribaucour  cut  the 
middle  surface  in  a  conjugate  system. 

Guichard  *  proved  that  this  property  is  characteristic  of  congru 
ences  of  Ribaucour.  In  order  to  obtain  this  result,  we  differentiate 
the  first  of  equations  (54)  with  respect  to  v,  and  in  the  reduction 
make  use  of  the  fact  that  X  and  p  satisfy  equations  (V,  22  )  and 
(53)  respectively.  This  gives 

/isyy*.    is  log  p   fi2\as. 

dv         \\  }  /  du     \    iu 


From  this  and  similar  equations  in  yv  and  zv  it  follows  that  a 
necessary  and  sufficient  condition  that  the  parametric  curves  form 
a  conjugate  system  is      ^   f!2V      d  T12V 
du\  1 S     to\ 2  J  ' 

When  this  condition  is  satisfied  by  a  system  of  curves  on  the 
sphere,  they  represent  the  asymptotic  lines  on  a  unique  surface  S, 
whose  coordinates  are  given  by  the  quadratures  (VI,  14) 


*Annales  L'Ecole  Nonnale,  Ser.  3,  Vol.  VI  (1889),  pp.  344,  345. 


422  RECTILINEAR  CONGRUENCES 

and  similar  expressions  for  y  and  z.     Combining  these  equations 

with  (54),  we  find  that 

dx  _  _  ^\  3xl  dx     ^  dxl  dx      ~  ^\  dxv  dx 

du=  Z  du  dv     **  'dv'du  =  ^~dv~dv  = 

Hence  S  and  Si  correspond  with  orthogonality  of  linear  elements, 
and  the  normals  to  the  former  are  parallel  to  the  lines  of  the  con 
gruence.  Hence  : 

A  necessary  and  sufficient  condition  that  the  developables  of  a 
congruence  cut  the  middle  surface  in  a  conjugate  system  is  that  their 
representation  be  that  also  of  the  asymptotic  lines  of  a  surface,  in  which 
case  the  latter  and  the  middle  surface  correspond  with  orthogonality 
of  linear  elements. 

EXAMPLES 

1.  When  the  coordinates  of  the  unit  sphere  are  in  the  form  (III,  35),  the  para 
metric  curves  are  asymptotic  lines.    Find  the  IF-congruences  for  which  the  sphere 
is  one  of  the  focal  sheets. 

2.  Let  vi  =fi(u)  +  0i  (w),  where  /;  and  </>t-  are  functions  of  u  and  u  respectively, 
and  i  =  1,  2,  3,  be  three  solutions  of  the  first  of  equations  (09),  in  which  case  X  =  0, 
and  let  61  in  (74)  be  unity.    Show  that  for  the  corresponding  ^-congruence  the  mid 
dle  surface  is  a  surface  of  translation  with  the  generatrices  u  =  const.,  v  =  const., 
that  the  functions  /t-  and  0,-  are  proportional  to  the  direction-cosines  of  the  binor- 
inals  to  these  generatrices,  and  that  the  intersections  of  the  osculating  planes  of 
these  generatrices  are  the  lines  of  the  congruence. 

3.  Show  that  isotropic  congruences  and  congruences  of  Guichard  are  congru 
ences  of  Ribaucour. 

4.  A  necessary  and  sufficient  condition  that  a  congruence  of  Ribaucour  be  nor 
mal  is  that  the  spherical  representation  of  its  developables  be  isothermic. 

5.  The  normals  to  quadrics  and  to  the  cyclidesof  Dupin  constitute  congruences 
of  Ribaucour. 

6.  When  the  middle  surface  of  a  congruence  is  plane,  the  congruence  is  of  the 
Ribaucour  type. 

7.  Show  that  the  congruence  of  Ribaucour,  whose  director  surface  is  a  skew 
helicoid,  is  a  normal  congruence,  and  that  the  normal  surfaces  are  molding  surfaces. 

8.  Show  that  a  necessary  and  sufficient  condition  that  a  congruence  of  Ribaucour 
be  normal  is  that  the  director  surface  be  minimal. 

GENERAL  EXAMPLES 

1  .  Through  each  line  of  a  congruence  there  pass  two  ruled  surfaces  of  the  con 
gruence  whose  lines  of  striction  lie  on  the  middle  surface  ;  their  equation  is 

edu*  +  (f  +  f')dudv 


they  are  called  the  mean  ruled  surfaces  of  the  congruence. 


GENERAL  EXAMPLES  423 

2.  Show  that  the  mean  ruled  surfaces  of  a  congruence  are  represented  on  the 
sphere  by  an  orthogonal  system  of  real  lines,  and  that  their  central  planes  (§  105) 
bisect  the  angles  between  the  focal  planes.    Let  u  =  const.  ,  v  =  const,  be  the  mean 
ruled  surfaces  and  develop  a  theory  analogous  to  that  in  §  167. 

3.  If  the  two  focal  surfaces  of  a  congruence  intersect,  the  intersection  is  the 
envelope  of  the  edges  of  regression  of  the  two  families  of  developable  surfaces  of 
the  congruence. 

4.  If  a  congruence  consists  of  the  lines  joining  points  on  two  twisted  curves,  the 
focal  planes  for  a  line  of  the  congruence  are  determined  by  the  line  and  the  tangent 
to  each  curve  at  the  point  where  the  curve  is  met  by  the  line. 

5.  In  order  that  the  lines  which  join  the  centers  of  geodesic  curvature  of  the 
curves  of  an  orthogonal  system  on  a  surface  shall  form  a  normal  congruence,  it  is 
necessary  and  sufficient  that  the  corresponding  radii  of  geodesic  curvature  be  func 
tions  of  one  another,  or  that  the  curves  in  one  family  have  constant  geodesic  curvature. 

6.  Let  S  be  a  surface  whose  lines  of  curvature  in  one  system  are  circles;  let  C 
denote  the  vertex  of  the  cone  circumscribing  S  along  a  circle,  and  L  the  corre 
sponding  generator  of  the  envelope  of  the  planes  of  the  circles  ;  a  necessary  and 
sufficient  condition  that  the  lines  through  the  points  C  and  the  corresponding  lines  L 
form  a  normal  congruence  is  that  the  distance  from  C  to  the  points  of  the  correspond 
ing  circle  shall  be  the  same  for  every  circle  ;  if  this  distance  be  denoted  by  a,  the 
radius  of  the  sphere  is  given  by    ™  _  p/2/  jn  \  a2\ 

where  the  accent  indicates  differentiation  with  respect  to  the  arc  of  the  curve  of 
centers  of  the  spheres. 

7.  Let  -S  be  a  surface  referred  to  its  lines  of  curvature,  Ci  and  C2  the  centers 
of  principal  normal  curvature  at  a  point,  GI  and  G2  the  centers  of  geodesic  curva 
ture  of  the  lines  of  curvature  at  this  point;  a  necessary  and  sufficient  condition 
that  the  line  joining  C2  and  G\  form  a  normal  congruence  is  that  p2  be  a  function 
of  Pgu,  or  that  one  of  these  radii  be  a  constant. 

8.  Let  S  be  a  surface  of  the  kind  defined  in  Ex.  6;  the  cone  formed  by  the 
normals  to  the  surface  at  points  of  a  circle  A  is  tangent  to  the  second  sheet  of  the 
evolute  of  -S  in  a  conic  T  (cf.  §  132).   Show  that  the  lines  through  points  of  T  and 
the  vertex  C  of  the  cone  which  circumscribes  8  along  A  generate  a  normal  con 
gruence,  and  that  C  lies  in  the  plane  of  F. 

9.  Given  an  isothermal  orthogonal  system  on  the  sphere  for  which  the  linear 
element  is  Z  _          *  +  cto2)  ; 


on  each  tangent  to  a  curve  v  =  const,  lay  off  the  segment  of  length  X  measured  from 
the  point  of  contact,  and  through  the  extremity  of  the  segment  draw  a  line  parallel  to 
the  radius  of  the  sphere  at  the  point  of  contact  .  Show  that  this  congruence  is  iso  tropic. 
10.  When  a  congruence  is  iso  tropic  and  its  direction-cosines  are  of  the  form 
(III,  35),  equation  (53)  reduces  to 


8uBv          (1-f-ww) 
Show  that  the  general  integral  is 

p  =  2  O0(v)  -  vf(u)](l  +  uv) 
where  /  and  0  are  arbitrary  functions  of  u  and  v  respectively.   Find  the  equations 
of  the  middle  surface. 


424  RECTILINEAR  CONGRUENCES 

11.  Show  that  the  intersections  of  the  planes 

(1  -  M2)x  -f  i  (1  +  w2)  y  +  2  uz  +  4/(w)  =  0, 
(1-  v2)z  -  i(l  +  vz)y  +  2vz  +40(u)=  0 

constitute  an  isotropic  congruence,  for  which  these  are  the  focal  planes  ;  that  the 
locus  of  the  mid-points  of  the  lines  joining  points  on  the  edges  of  regression  of  the 
developables  enveloped  by  these  planes  is  the  minimal  surface  which  is  the  middle 
envelope  of  the  congruence,  by  rinding  the  coordinates  of  the  point  in  which  the 
tangent  plane  to  this  surface  meets  the  intersection  of  the  above  planes. 

12.  Show  that  the  middle  surface  of  an  isotropic  congruence  is  the  most  general 
surface  which  corresponds  to  a  sphere  with  orthogonality  of  linear  elements,  and 
that  the  corresponding  associate  surface  in  the  infinitesimal  deformation  of  the 
sphere  is  the  minimal  surface  adjoint  to  the  middle  envelope. 

13.  Find  the  surface  associate  to  the  middle  surface  of  an  isotropic  congruence 
when  the  surface  corresponding  to  the  latter  with  orthogonality  of  linear  elements 
is  a  sphere,  and  show  that  it  is  the  polar  reciprocal,  "with  respect  to  the  imaginary 
sphere  x2  -f  y2  -f  z2  +  1  =  0,  of  the  minimal  surface  adjoint  to  the  middle  envelope 
of  the  congruence. 

14.  The  lines  of  intersection  of  the  osculating  planes  of  the  generatrices  of  a 
surface  of  translation  constitute  a  IT-congruence  of  which  the  given  surface  is  the 
middle  surface  ;  if  the  generatrices  be  curves  of  constant  torsion,  equal  but  of 
opposite  sign,  the  congruence  is  normal  to  a  TF-surface  of  the  type  (VIII,  72). 

15.  If  the  points  of  a  surface  S  be  projected  orthogonally  upon  any  plane  A, 
and  if,  after  the  latter  has  been  rotated  about  any  line  normal  to  it  through  a 
right  angle,  lines  be  drawn  through  points  of  A  parallel  to  the  corresponding  nor 
mals  to  -S,  these  lines  form  a  congruence  of  Ribaucour. 

16.  A  necessary  and  sufficient  condition  that  the  tangents  to  the  curves  v  —  const. 
on  a  surface,  whose  point  equation  is  (VI,  26),  shall  form  a  congruence  of  Ribaucour  is 

aa_S6 


du      dv        dudv 

17.  Show  that  the  tangents  to  each  system  of  parametric*  curves  on  a  surface 
form  congruences  of  Ribaucour  when  the  point  equation  is 


where  Ui  and  V\  are  functions  of  u  and  v  respectively,  and  the  accents  indicate 
differentiation. 

18.  Show  that  if  the  parametric  curves  on  a  surface  S  form  a  conjugate  system, 
and  the  tangents  to  the  curves  of  each  family  form  a  congruence  of  Ribaucour,  the 
same  is  true  of  the  surfaces  Si  and  S_i,  which  together  with  S  constitute  the  focal 
surfaces  of  the  two  congruences. 

19.  Show  that  the  parameter  of  distribution  p  of  the  ruled  surface  of  a  con 
gruence,  determined  by  a  value  of  dv/du,  is  given  by 

1 


P  = 


-f 
e  du  +  /du,      f'du  -f  g  dv 


GENERAL  EXAMPLES  425 

20.  Show  that  the  mean  ruled  surfaces  (cf.  Ex.  1)  of  a  congruence  are  char 
acterized  by  the  property  that  for  these  surfaces  the  parameter  of  distribution 
has  the  maximum  and  minimum  values. 

21.  If  S  and  SQ  are  two  associate  surfaces,  and  through  each  point  of  one  a  line 
be  drawn  parallel  to  the  corresponding  radius  vector  of  the  other,  the  developables  of 
the  congruence  thus  formed  correspond  to  the  common  conjugate  system  of  S  and  SQ, 

22.  In  order  that  two  surfaces  S  and  SQ  corresponding  with  parallelism  of 
tangent  planes  be  associate  surfaces,  it  is  necessary  and  sufficient  that  for  the 
congruence  formed  by  the  joins  of  corresponding  points  M  and  MQ  of  these  sur 
faces  the  developables  cut  S  and  SQ  in  their  common  conjugate  system,  and  that 
the  focal  points  M  and  MQ  form  a  harmonic  range. 

23.  In  order  that  a  surface  S  be  iso thermic,  it  is  necessary  and  sufficient  that 
there  exist  a  congruence  of  Ribaucour  of  which  S  is  the  middle  surface,  such  that 
the  developables  cut  S  in  its  lines  of  curvature. 


CHAPTER  XIII 

CYCLIC  SYSTEMS 

174.  General  equations  of  cyclic  systems.  The  term  congruence 
is  not  restricted  to  two-parameter  systems  of  straight  lines,  but  is 
applied  to  two-parameter  systems  of  any  kind  of  curves.  Darboux  * 
has  made  a  study  of  these  general  congruences  and  Ribaucourf  has 
considered  congruences  of  plane  curves.  Of  particular  interest  is 
the  case  where  these  curves  are  circles.  Ribaucour  has  given  the 
name  cyclic  systems  to  congruences  of  circles  which  admit  of  a  one- 
parameter  family  of  orthogonal  surfaces.  This  chapter  is  devoted 
to  a  study  of  cyclic  systems. 

We  begin  with  the  general  case  where  the  planes  of  the  circles 
envelop  a  nondevelopable  surface  S.  We  associate  with  the  latter 
a  moving  trihedral  (§  68),  and  for  the  present  assume  that  the 
parametric  curves  on  the  surface  are  any  whatever. 

As  the  circles  lie  in  the  tangent  planes  to  S,  the  coordinates 
of  a  point  on  one  of  them  with  respect  to  the  corresponding 
trihedral  are  of  the  form 

(1)  a  +  Rcos0,     b+Rsm0,     0, 

where  a,  b  are  the  coordinates  of  the  center,  R  the  radius,  and  0 
the  angle  which  the  latter  to  a  given  point  makes  with  the  moving 


In  §  69  we  found  the  following  expressions  for  the  projections 
of  a  displacement  of  a  point  with  respect  to  the  moving  axes  : 

(dx+%du  +  ^dv  +  (qdu  +  q^v)  z  —  (rdu  +  rtdv)  y, 

(")  \  dy  +  77  du  4-  rj^dv  +  (rdu  +  rvdv]  x  —  (pdu  +  p^dv)  z, 

\dz  +(p  du  +p1dv)  y  —  (qdu+  qvdv)  x, 


*  Vol.  II,  pp.  1-10;  also  Eisenhart,  Congruences  of  Curves,  Transactions  of  the  Amer. 
Math.  Soc.,  Vol.  IV  (1903),  pp.  470-488. 

t  Memoire  sur  la  theorie  generale  des  surfaces  courbes,  Journal  des  Mathtmatiques, 
Ser.  4,  Vol.  VII  (1891),  §  117  et.  seq. 

426 


GENERAL  EQUATIONS 


427 


where  the  translations  f ,  f  1?  T;,  ^  and  the  rotations  p,q,r->  p^  qv,  r^ 
satisfy  the  conditions 

'dp_d£i_     __        d|__Mi  =     — 


(3) 


When    the    values    (1)   are   substituted   in   (2)   the   latter    are 
reducible   to 


dr      dr. 

—  ^Wl— «PH      Ph-M^fl- 


J  du  +  J^v  -|-  cos  QdR  —  (dd  +  rdu+  r^dv)  R  sin  ^, 
#  ^w  +  ^jrfu  +  sin  6dR  +  (dd+rdu  +  r^dv]  R  cos  6, 
(p  du  -f  ^^v)  (b  +  R  sin  0)  —  (y  du  +  q^dv)  (a+R  cos  0 
where  we  have  put,  for  the  sake  of  brevity, 


(5) 


^ 

du 


The  conditions  that 

—  (—\  =  —  (^ 

du  \dv]      dv  \du/  '  du  \dv 

are  reducible,  by  means  of  (3),  to 


\du 


(6) 


_ 

dv       du 


The  direction-cosines  of  the  tangent  to  the  given  circle  at  the 
point  (1)  are 
(7)  —  sin0,     cos0,     0. 

Hence  the  condition  that  the  locus  of  the  point,  as  u  and  v  vary, 
be  orthogonal  to  the  circle  is  that  the  sum  of  the  expressions  (4) 
multiplied  respectively  by  the  quantities  (7)  be  zero.  This  gives 


428  CYCLIC  SYSTEMS 

In  order  that  the  system  of  circles  be  normal  to  a  family  of  sur 
faces  this  equation  must  admit  of  a  solution  involving  a  parameter. 
Since  it  is  of  the  form 


(9) 

the  condition  that  such  an  integral  exist  is  that  the  equation 


be    satisfied    identically.  *     For    equation    (8)    this    condition   is 
reducible  to 


In  order  that  this  equation  be  satisfied  identically,  the  expressions 
in  the  brackets  must  be  zero.  If  they  are  not  zero,  it  is  possible 
that  the  two  solutions  of  this  equation  will  satisfy  (8),  and  thus 
determine  two  surfaces  orthogonal  to  the  congruence  of  circles. 
Hence  we  have  the  theorem  of  Ribaucour: 

If  the  circles  of  a  congruence  are  normal  to  more  than  two  surfaces, 
they  form  a  cyclic  system. 

The   equations   of   condition   that   the    system   be   cyclic   are 

consequently 

dR    .      dR   . 


The  total  curvature  of  S  is  given  by  (cf.  §  70) 


*  Murray,  Differential  Equations,  p.  137.   New  York,  1897;  also  Forsyth,  Differential 
Equations,  p.  257.   London,  1888. 


THEOREMS  OF  RIBAUCOUR 


429 


From  this  and  (5)  it  is  seen  that  equations  (12)  involve  only 
functions  relating  to  the  linear  element  of  S  and  to  the  circle. 
Hence  we  have  the  theorem  of  Ribaucour: 

If  the  envelope  of  the  planes  of  the  circles  of  a  cyclic  system  be 
deformed  in  any  manner  without  disturbing  the  size  or  position  of 
the  circles  relative  to  the  point  of  contact,  the  congruence  of  circles 
continues  to  form  a  cyclic  system. 

Furthermore,  if  we  put  Q 

t  =  tan  —  > 

z* 

equation  (8)  assumes  the  Riccati  form, 

dt  +  (af  +a2t  +  a3)  du  +  (b/  +  bjt  +  b3)  dv  =  0, 

where  the  #'s  and  5's  are  functions  of  u  and  v.  Recalling  a  funda 
mental  property  of  such  equations  (§  14),  we  have  : 

Any  four  orthogonal  surfaces  of  a  cyclic  system  meet  the  circles  in 
four  points  whose  cross-ratio  is  constant. 

Since  by  hypothesis  S  is  nondevelopable,  equations  (12)  may 
be  replaced  by 


(13) 


— 

du 

^ 

dv 


AB,  - 


-  trf)  JBT  =  0. 
By  (5)  the  first  two  of  these  equations  are  reducible  to 


(14) 


du 

a 


The  condition  of  integrability  of  these  equations  is 

(15)  ^{  +  g,-^f1-g,l-r(f^-J«1)-r1(«,-^. 

dv        cv         du         cu 

Instead  of  considering  this  equation,  we  introduce  a  function  <j> 
by  the  equation 

(16)  24>=,K2-a2-&2, 


430  CYCLIC  SYSTEMS 

and  determine  the  condition  which  </>  must  satisfy.    We  take  for 
a  and  b  the  expressions  obtained  by  solving  (14);  that  is 


(17) 


Now  the  equation  (15)  vanishes  identically,  and  the  only  other 
condition  to  be  satisfied  is  the  last  of  (13);  this,  by  the  substi 
tution  of  these  values  of  a,  6,  R,  becomes  a  partial  differential 
equation  in  <£  of  the  form 

(18)        £-r  ^_r  _(__r_]  -\-J^  +  L — —  +  M— ~  +  N=  0, 

du    dv        \dudv  /         du  ducv          du 

where  «7,  X,  JHf,  JV  denote  functions  of  <£>,  f ,  •  •  •  rt,  and  their  deriva 
tives  of  the  first  order.  Conversely,  each  solution  of  this  equation 
gives  a  cyclic  system  whose  circles  lie  in  the  tangent  planes  to  S. 

EXAMPLES 

1.  Let  S  be  a  surface  of  revolution  defined  by  (III,  99),  and  let  Tbe  the  trihedral 
whose  x-axis  is  tangent  to  the  curve  v  =  const.    Determine  the  condition  which  the 
function  \f/  (u)  must  satisfy  in  order  that  the  quantities  a,  b  in  (1)  may  have  the  values 

a  =  _*w_.     6  =  l, 

and  determine  also  the  expression  for  R. 

2.  A  necessary  and  sufficient  condition  that  all  the  circles  of  a  cyclic  system 
whose  planes  envelop  a  nondevelopable  surface  shall  have  the  same  radius,  is  that 
the  planes  of  the  circles  touch  their  envelope  S  at  the  centers  of  the  circles,  and 
that  S  be  pseudospherical. 

3.  Let  S  be  a  surface  referred  to  an  orthogonal  system  of  lines,  and  let  T  be 
the  trihedral  whose  z-axis  is  tangent  to  the  curve  v  =  const.    With  reference  to  the 
trihedral  the  equations  of  a  curve  in  the  tangent  plane  are  of  the  form 

x  =  p  cos  0,         y  =  p  sin  0,        z  =  0, 

where  in  general  p  is  a  function  of  0,  w,  and  v.  Show  that  the  condition  that  there 
be  a  surface  orthogonal  to  these  curves  is  that  there  exist  a  relation  between  0,  u, 
and  v  which  satisfies  the  equation 


U  sin  0  —  -f  prji  cos  0 
30 

When  this  condition  is  satisfied  by  a  function  0  which  involves  an  arbitrary  con 
stant,  there  is  an  infinity  of  normal  surfaces.  In  this  case  the  curves  are  said  to 
form  a  normal  congruence. 

4.  When  the  surface  enveloped  by  the  planes  of  the  curves  of  a  normal  con 
gruence  of  plane  curves  is  deformed  in  such  a  way  that  the  curves  remain  invari 
ably  fixed  to  the  surface,  the  congruence  continues  to  be  normal. 


CYCLIC  CONGRUENCES  431 

175.  Cyclic  congruences.  The  axes  of  the  circles  of  a  cyclic  sys 
tem  constitute  a  rectilinear  congruence  which  Bianchi  *  has  called 
a  cyclic  congruence.  In  order  to  derive  the  properties  of  this  con 
gruence  and  further  results  concerning  cyclic  systems,  we  assume 
that  the  parametric  curves  on  S  correspond  to  the  developables  of 
the  congruence. 

The  coordinates  of  the  focal  points  of  a  line  of  the  congruence 
with  reference  to  the  corresponding  trihedral  are  of  the  form 
a,  ft,  p^  a,  ft,  />2.  On  the  hypothesis  that  the  former  are  the 
coordinates  of  the  focal  point  for  the  developable  v  =  const, 
through  the  line,  we  have,  from  (2), 

-  +  ?  +?/°i—  rft  =  0,  -  +  T)—ppl+ra  =  Q. 

cu  cu 

Proceeding  in  like  manner  with  the  other  point,  we  obtain  a  pair 
of  similar  equations.  All  of  these  equations  may  be  written  in  the 
abbreviated  form 

(19)  A  +  qPl=0,     Ji-PPl=Q,     ^+?lft=0,     J?,-^p2=0, 

in  consequence  of  (5).  When  these  values  are  substituted  in  the 
last  of  equations  (13),  it  is  found  that 

(20)  Sf=-P1pf 

Hence  the  lines  joining  a  point  on  the  circle  to  the  focal  points  are 
perpendicular.  If  we  put 


thus  indicating  by  2  p  the  distance  between  the  focal  points,  and  by 
8  the  distance  between  the  center  of  the  circle  and  the  mid-point 
of  the  line  of  the  congruence,  we  find  that 

We  replace  this  equation  by  the  two 

(21)  8  —  p  cos  <r,         R  —  p  sin  cr, 

thus  defining  a  function  a-.    Now  we  have 

/51=/E)(coso-+l),          /?2=/o(cos<r  —  1), 
so  that  equations  (19)  may  be  written 
A  =  —  o. 


(22) 

^  *i  =  _  qip  (cos  o-  —  1),       Bl  =  prf  (cos  o-  —  1). 

*Vol.  II,  p.  161. 


432  CYCLIC  SYSTEMS 

By  means  of  (5)  equation  (15)  can  be  put  in  the  form 


When  the  values  (22)  are  substituted  in  this  equation,  it  becomes 


Since  by  (3)  the  expression  in  the  first  parenthesis  is  zero,  the  same 
is  true  of  the  second,  and  so  we  have 


But  these  are  the  conditions  (V,  67)  that  the  parametric  curves 
on  S  form  a  conjugate  system.  Hence  we  have  the  theorem  of 
Ribaucour  : 

On  the  envelope  of  the  planes  of  the  circles  of  a  cyclic  system  the 
curves  corresponding  to  the  developables  of  the  associated  cyclic  con 
gruence  form  a  conjugate  system. 

176.  Spherical  representation  of  cyclic  congruences.  When  the 
expressions  (22)  are  substituted  in  (6),  we  obtain 


do      da. 


dp      dp, 


Since  pql  —  plq  =t=  0  unless  Sis  developable,  the  preceding  equations 
may  be  replaced  by 

£[/3(coso--l)]=2/0{122}VH-^), 

d  _  /12V 

—  [/3(cos  a  -hi)]  =  —  2ps        \ +  (a(l\—°P\)'i 
cv  L  1  J 

where  the  Christoff el  symbols  are  formed  with  respect  to 

(24)  (pdu+pldv)2-\-(qdu-{-qldvY, 

the  linear  element  of  the  spherical  representation  of  S. 


SPHERICAL  REPRESENTATION  433 

When  in  like  manner  we  substitute  in  the  first  two  of  equations 
J?2  =  p2  sin2  cr  =  p2  (1  —  cos2  o-), 


(13),  taking 
we  obtain 


.  dp        p  cos  o-     d  , 

—  cos  a-)  —  —  ---         -  —  cos  a  =pb  —  qa, 


du      1 4-  cos  a  du 


n  ,     \  p 
(1  +  cos  o)  -J-  — 

7 


P  cos 


1  —  cos  o- 
From  these  equations  and  (23)  we  find 


*  r 

—  cos  a  —  — p  o 


(25) 

d 

—  cos 

dv 

a  =  2  (cos  a 

1J12V 

)\~i   i  ' 
\~  1.  J 

and 

(cos<r  —  1)^ 

=  —  2  p  cos 

ri2V 

oj        r  4~  ^<^  —  pO) 

(26) 

(cos«r+l)|; 

=  —  2  p  cos 

J12V 

°"\1  J  "^^"^  ' 

The  condition  of  integrability  of  equations  (25)  is  reducible  to 


If  the  expression  for  cos  a-  obtained  from  this  equation  be  substi 
tuted  in  (25),  we  find  two  conditions  upon  the  curves  on  the  sphere 
in  order  that  they  may  represent  the  developables  of  a  cyclic  con 
gruence.  A  particular  case  is  that  in  which  (27)  is  identically  sat 
isfied,  when  the  two  conditions  are 


(28) 


ri  O"\ '      ^   n  O"^  '         n  o^  /  f  1  o"\  / 
121      1/121  _  2  f  121  f  121 

ll  J       0v  I  2  J  "     ll  J  12  /  ' 


It  is  now  our  purpose  to  show  that  if  any  system  of  curves  on 
the  sphere  satisfies  either  set  of  conditions,  all  the  congruences 
whose  developables  are  thus  represented  on  the  sphere  are  cyclic. 

We  assume  that  the  sphere  is  referred  to  such  a  system  and  that 
we  have  a  solution  p  of 


434  CYCLIC  SYSTEMS 

By  the  method  of  §  167,  or  that  hereinafter  explained,  we  find  the 
middle  surface  of  the  congruence.  Then  we  take  the  point  on  each 
line  at  the  distance  —  p  cos  a-  from  the  mid-point  as  the  center  of  the 
circle  of  radius  p  sin  a  and  for  which  the  line  is  the  axis.  These  cir 
cles  form  a  cyclic  system,  as  we  shall  show. 

In  the  first  place  we  determine  the  middle  surface  with  reference 
to  a  trihedral  of  fixed  vertex,  whose  2-axis  coincides  with  the  radius 
of  the  sphere  parallel  to  the  line  of  the  congruence  and  whose  x- 
and  #-axes  are  any  whatever.  If  #0,  #0,  z0  denote  the  coordinates 
of  the  mid-point  of  a  line  with  reference  to  the  corresponding  tri 
hedral,  the  coordinates  of  the  focal  points  are 


From  (2)  it  is  seen  that  if  these  points  correspond  to  the  develop- 
ables  v  =  const,  and  u  =  const,  respectively,  we  must  have 


Since  pql  —  plq  =t=  0,  the  conditions  of  integrability  of  these  equa 
tions  can  be  put  in  the  form 


(30) 


It  is  readily  found  that  the  condition  of  integrability  of  these  equa 
tions  is  reducible  to  (29). 

It  will  be  to  our  advantage  to  have  also  the  coordinates  of  the 
point  of  contact  of  the  plane  of  the  circle  with  its  envelope  S.  If 
x,  y,  ZQ  —  p  cos  a  denote  these  coordinates  with  reference  to  the 
above  trihedral,  it  follows  from  (2)  that 

—  (z0-  p  cos  v)  +  py-qx  =  0, 

o 

—  (z0  -  p  cos  a)  +  p^y  -  q,x  =  0. 


CYCLIC  CONGRUENCES  435 

If  these  equations  be  subtracted  from  the  respective  ones  of  (30), 
the  results  are  reducible,  by  means  of  (25),  to 

(cos  a  -1)  -£  +  2  p  cos  a-  1  g  j  +  p(yQ-  y)  -  q(x0-  x)  =  0, 
(cos  cr  +  1)       +  2  p  cos  a-  V^-  y)-  ftfo-  x)  =  0, 


which  are  the  same  as  (26).  For,  the  quantities  x0  —  x,  yQ  —  y  are 
the  coordinates  of  the  center  of  the  circle  with  reference  to  the  tri 
hedral  parallel  to  the  preceding  one  and  with  the  corresponding 
point  on  S  for  vertex. 

If,  then,  we  have  a  solution  cr  of  (25)  and  p  of  (29),  the  corre 
sponding  values  of  a  and  b  given  by  (26)  satisfy  (22),  since  the 
latter  are  the  conditions  that  the  parametric  curves  on  the  sphere 
represent  the  developables  of  the  congruence.  However,  we  have 
seen  that  when  the  values  (22)  are  substituted  in  (12),  we  obtain 
equations  reducible  to  (25)  and  (26).  Hence  the  circles  constructed 
as  indicated  above  form  a  cyclic  system. 

Since  equations  (25)  admit  only  one  solution  (27)  unless  the  con 
dition  (28)  is  satisfied,  we  have  the  theorem: 

With  each  cyclic  congruence  there  is  associated  a  unique  cyclic  sys 
tem  unless  it  is  at  the  same  time  a  congruence  of  Ribaucour,  in  which 
case  there  is  an  infinity  of  associated  cyclic  systems. 

Recalling  the  results  of  §  141,  we  have  the  theorem  of  Bianchi  *  : 

When  the  total  curvature  of  a  surface  referred  to  its  asymptotic 
lines  is  of  the  form  - 


~  [* 

it  is  the  surface  generatrix  of  a  congruence  of  Ribaucour  which  is 
cyclic  in  an  infinity  of  ways,  and  these  are  the  only  cyclic  congru 
ences  with  an  infinity  of  associated  cyclic  systems. 

In  this  case  the  general  solution  of  equations  (25)  is 

(31)  cos  <,=>-*  +  *, 

<#>  +  ^ 

where  a  is  an  arbitrary  constant. 

*  Vol.  II,  p.  165. 


436  CYCLIC  SYSTEMS 

177.  Surfaces  orthogonal  to  a  cyclic  system.  In  this  section  we 
consider  the  surfaces  Sl  orthogonal  to  the  circles  of  a  cyclic  sys 
tem.  Since  the  direction-cosines  of  the  normals  to  the  surfaces 
with  reference  to  the  moving  trihedral  in  §  174  are  —  sin  6,  cos  0,  0, 
the  spherical  representation  of  these  surfaces  is  given  by  the  point 
whose  coordinates  are  these  with  respect  to  a  trihedral  of  fixed 
vertex  parallel  to  the  above  trihedral.  From  (2)  we  find  that  the 
expressions  for  the  projections  of  a  displacement  of  this  point  are 

—  cos  0(d0  +  rdu  +  ^  dv), 

—  sin  6  (dd  +  r  du  +  r1  dv), 

(p  du  +  p^dv)  cos  6  -f  (q  du  +  qvdv)  sin  0. 

Moreover,  by  means  of  (8),  (21),  (22),  we  obtain  the  identity 

(32)  sin  a- (dd  +  r  du  +  r^dv)  =  —  (1 4-  cos  a) ( p  cos  6  -f  q  sin  6)  du 

+  (1  —  cos  a)  ( pl  cos  6  +  <?!  sin  0)  dv. 

Hence  the  linear  element  of  the  spherical  representation  of  ^  is 

(33)  da*=  T — — -  (p  cosO  +  q  sin  0fdu* 

-L  — •  COS  O~ 


1\  r   1 
+  COSO- 

Since  the  parametric  curves  on  the  sphere  form  an  orthogonal 
system,  the  parametric  curves  on  the  surface  are  the  lines  of 
curvature,  if  they  form  an  orthogonal  system.  In  order  to  show 
that  this  condition  is  satisfied,  we  first  reduce  vthe  expressions  (4) 
for  the  projections  of  a  displacement  of  a  point  on  Sv  by  means 
of  (21),  (22),  (25),  (26),  and  (32),  to 


(34) 


Cdu  Ddv     \ 

cos  v  sin  <r  (  —  —  —  —       )» 

,1  —  cos  a      1  +  cos  07 

,      Cdu  Ddv    \ 

sin  0  sin  <T  i 


—  cos  <r      1  +  cos  cr, 
Cdu  +  Ddv, 
where  we  have  put 


=  pi(b  +  R  sin  0)—q1(a+E  cos  0). 


NORMAL  CYCLIC  CONGRUENCES  437 

Hence  the  linear  element  of  S1  is 

(36)  ds*  =  2  .  °  du     +  2    D    V    , 

1  —  COS  (T  1  +  COS  cr 

from  which  it  is  seen  that  the  parametric  curves  on  Sl  form  an 
orthogonal  system,  and  consequently  are  the  lines  of  curvature. 

Furthermore,  it  is  seen  from  (34)  that  the  tangents  to  the  curves 
v  =  const.,  u  =  const,  make  with  the  plane  of  the  circle  the  respec 
tive  angles 

.„-.  i/l  —  COScrX  ./      l-fCOSCT\ 

(37)  tan"1  — : ),        tan"1! • 

\    sin  cr    /  \         sin  a    / 

But  it  follows  from  (21)  that  the  lines  joining  a  point  on  the  cir 
cumference  of  a  circle  to  the  focal  points  of  its  axis  make  the 
angles  (37)  with  the  radius  to  the  point.  Hence  we  have : 

The  lines  of  curvature  on  a  surface  orthogonal  to  a  cyclic  system 
correspond  to  the  developables  of  the  congruence  of  axes  of  the  circles, 
and  the  tangents  to  the  two  lines  of  curvature  through  a  point  of  the 
surface  meet  the  corresponding  axis  in  its  focal  points. 

178.  Normal  cyclic  congruences.  Since  the  developables  of  a 
cyclic  congruence  correspond  to  a  conjugate  system  on  the  enve 
lope  S  of  the  planes  of  the  circles,  this  system  consists  of  the 
lines  of  curvature  when  the  congruence  is  normal,  and  only  in 
this  case  (cf.  §  83).  If,  under  these  conditions,  we  take  two  of  the 
edges  of  the  trihedral  tangent  to  the  lines  of  curvature,  we  have 

and  equations  (25)  become 

d  -,  03,  d  ,         .     cr       d 


By  a  suitable  choice  of  parameters  we  have 


"  «tt 


so  that  if  we  put  &>  =  —  cr/2,  the  linear  element  of  the  sphere  is 
(39)  d(r*=  sinW%2+  cosWv2. 

Comparing  this  result  with  (§  119),  we  have  the  theorem: 

The  normals  to  a  surface  2  with  the  same  spherical  representation 
of  its  lines  of  curvature  as  a  pseudospherical  surface  constitute  the 
only  kind  of  normal  cyclic  congruences. 


438  CYCLIC  SYSTEMS 

Since  the  surface  2  and  the  envelope  £  of  the  planes  of  the 
circles  have  the  same  representation  of  their  lines  of  curvature, 
the  tangents  to  the  latter  at  corresponding  points  on  the  two 
surfaces  are  parallel.  Hence  with  reference  to  a  trihedral  for  2 
parallel  to  the  trihedral  for  S  the  coordinates  of  a  point  on  the 
circle  are  R  cos  0,  R  sin  6,  p,  where  ft  remains  to  be  determined 
and  6  is  given  by  (32),  which  can  be  put  in  the  form 

,*(\^  cd      d(D  .    n          dO      da)  .  a 

(40)  —  H =  cos  co  sm  0, 1 =  —  sin  &>  cos  6. 

du      dv  dv      du 

If  we  express,  by  means  of  (2),  the  condition  that  all  displace 
ments  of  this  point  be  orthogonal  to  the  line  whose  direction- 
cosines  are  —  sin  0,  cos  0,  0,  the  resulting  equation  is  reducible, 
by  means  of  (40),  to 

sin  9  (R  cos  o>  —  fi  sin  o>  —  f )  du 

—  cos  0  (R  sin  &>  -f-  //,  cos  &>  —  77^  dv  —  0. 

Hence  the  quantities  in  parentheses  are  zero,  from  which  we  obtain 

(41)  R  =  i;  cos  &)  -h  T]I  sin  w,         ^  =  —  f  sin  o>  +  rjl  cos  o>. 

When,  in  particular,  2  is  a  pseudospherical  surface  of  curvature 
-I/a2,  we  have  (VIII,  22) 

f  =  a  cos  &),          rjl  =  a  sin  &>, 

so  that  R=  a  and  /x  =  0.  Hence  the  circles  are  of  constant  radius 
and  the  envelope  of  their  planes  is  the  locus  of  their  centers 
(cf.  Ex.  2,  §  174).  Conversely,  when  the  latter  <  condition  is  sat 
isfied,  it  follows  from  (13)  that  R  is  constant.  Moreover,  in  this 
case  p^  and  p2,  as  defined  in  §  175,  are  the  principal  radii  of  the 
surface,  which  by  (20)  is  pseudospherical.  When  these  values 
are  substituted  in  (36)  and  (33),  it  is  found  that  the  linear  ele 
ment  of  each  orthogonal  surface  is 

ds*  =  a? (cos2 6 du* -f  sin2 6 dvz), 
and  of  its  spherical  representation 

(42)  d<r*=sm*0du*+  cos2<W. 

Hence  these  orthogonal  surfaces  are  the  transforms  of  2  by  means 
of  the  Bianchi  transformation  (§  119). 


PLANES  OF  CIRCLES  TANGENT  TO  A  CUKVE      439 

The  expression  (42)  is  the  linear  element  of  the  spherical  rep 
resentation  of  the  surfaces  orthogonal  to  the  circles  associated  with 
any  surface  2,  whether  it  be  pseudospherical  or  not,  whose  spherical 
representation  is  given  by  (39).  Since  these  orthogonal  surfaces 
have  this  representation  of  their  lines  of  curvature,  they  are  of  the 
same  kind  as  2.  We  have  thus  for  all  surfaces  with  the  same  rep 
resentation  of  their  lines  of  curvature  as  pseudospherical  surfaces, 
a  transformation  into  similar  surfaces  of  which  the  Bianchi  trans 
formation  is  a  particular  case  ;  we  call  it  a  generalized  Bianchi 
transformation.* 

179.  Cyclic  systems  for  which  the  envelope  of  the  planes  of  the 
circles  is  a  curve.  We  consider  now  the  particular  cases  which  have 
been  excluded  from  the  preceding  discussion,  and  begin  with  that 
for  which  the  envelope  S  of  the  planes  of  the  circles  is  a  curve  C. 

We  take  the  moving  trihedral  such  that  its  zy-plane,  as  before, 
is  that  of  the  circle,  and  take  the  z-axis  tangent  to  C.  If  s  denotes 
the  arc  of  the  latter,  we  have 

ds  =  f;  du  +  ^dv,         ??  =  rj1  =  0, 
and  by  (3) 

(43)  r^-r^  =  Q,         rfx-ftf  =  0. 

From  (14),  (15),  and  (16)  it  follows  that  a  and  </>  are  functions  of  s, 
so  that  these  equations  may  be  replaced  by 

(44)  .K2=a2+&2 


If  the  parametric  curves  on  the  sphere  represent  the  developables 
of  the  congruence,  the  conditions  (19)  must  hold.  But  from  (5), 
(15),  and  (43)  we  obtain  ^«.  _^  £  _  Q 

If  the  values  from  (19)  be  substituted  in  this  equation,  we  have, 
from  (43),  ^-^=0. 

Hence  the  focal  surfaces  coincide.    If  we  put 

P  =  Pi=P* 

in  (19)  and  substitute  in  the  last  of  (12),  we  obtain 
(^+^2)(^1-^)==0. 

*Cf.  American  Journal,  Vol.  XXVI  (1905),  pp.  127-132. 


440  CYCLIC  SYSTEMS 

The  vanishing  of  pql  —  plq  is  the  condition  that  there  be  a  single 
infinity  of  planes,  which  case  we  exclude  for  the  present.  Hence 
p  =  ±  iR  ;  that  is,  the  developables  of  the  cyclic  congruence  are 
imaginary. 

Instead  of  retaining  as  parametric  curves  those  representing  the 
developables,  we  make  the  following  choice.  We  take  the  arc  of  C 
for  the  parameter  u  ;  consequently  f  =1,  ^=0.  Since  77  =  ^x=  0 
also,  we  have,  from  (3), 

«.-'-«•    2MB-* 

hence  we  may  choose  the  parameter  v  so  that  p  =  0,  ^=1. 
From  (3)  it  follows,  furthermore,  that 

dq  dr 

i=T>     B-V=-* 

of  which  the  general  integral  is 

q=Ul  cos  v  +  U2  sin  v,          r  =  —  Ul  sin  v  -f  Uz  cos  v, 
where  U^  and  U2  are  arbitrary  functions  of  u.    From  (5)  we  have 

A  =  j>"(u)+\-rl,         ^=0,         A  =  7T' 

#y 

so  that  the  third  of  equations  (12)  is  reducible  by  (44)  to 

(*"+!)- 

cv  d  U  sin  v  —  U  cos  v 


Hence  if  we  take  for  a  any  function  of  u  denoted  by  <f>r(u),  equation 

(45)  gives  6,  and  R  follows  directly  from  (44). 

180.  Cyclic  systems  for  which  the  planes  of  the  circles  pass 
through  a  point.  If  the  planes  of  the  circles  of  a  cyclic  system 
pass  through  a  point  0,  we  take  it  for  the  origin  and  for  the 
vertex  of  a  moving  trihedral  whose  z-axis  is  parallel  to  the  axis 
of  the  circle  under  consideration.  In  this  case  equations  (14) 
may  be  replaced  by 

(46)  A>2  =  tf2  +  62-*, 

where  c  denotes  a  constant.    But  this  is  the  condition  that  all  the 
circles  are  orthogonal  to  a  sphere  with  center  at  0,  or  cut  it  in 


PLANES  OF  CIKCLES  THROUGH  A  POINT  441 

diametrically  opposite  points,  or  pass  through  0,  according  as  c  is 
positive,  negative,  or  zero.    Hence  we  have  the  theorem  : 

If  the  planes  of  the  circles  of  a  cyclic  system  pass  through  a  point, 
the  circles  are  orthogonal  to  a  sphere  with  its  center  at  the  point,  or 
meet  the  sphere  in  opposite  points,  or  pass  through  the  center. 

From  geometrical  considerations  we  see  that  the  converse  of  this 
theorem  is  true. 

When  c  in  (46)  is  zero  all  the  circles  pass  through  0.  Then 
by  (21)  we  have 

(47)  a  =  —  p  sin  <r  cos  6,         b  =  —  p  sin  a  sin  6, 

and  equations  (26)  become 


(cos  or  —  1)  —  ^£  =  —  2  cos  a-  \  0  f  +  sin  a-  (p  sin  6  —  q  cos 

, 

(cos  <r  +1)  -  S-L.  —  —.  2cos<rj       f  -\-smo-(plsm0  — 

These  equations  are  obtained  likewise  when  we  substitute  the 
values  (47)  in  equations  (22)  and  reduce  by  means  of  (25)  and  (32). 
Because  of  (22)  the  function  p  given  by  (26)  is  a  solution  of  (29), 
arid  therefore  p  given  by  (48)  is  a  solution.  But  the  solution  6  of 
(32)  involves  a  parameter.  Hence  we  have  the  theorem  of  Bianchi  *  : 
Among  all  the  cyclic  congruences  with  the  same  spherical  repre 
sentation  of  their  developables  there  are  an  infinity  for  which  the 
circles  of  the  associated  cyclic  system  pass  through  a  point. 

If  we  take  the  line  through  0  and  the  center  of  the  circle  for 
the  z-axis  of  the  trihedral,  equation  (11)  must  admit  of  the  solu 
tion  0  =  TT,  and  consequently  must  be  of  the  form 


In  order  that  this  equation  admit  of  a  solution  other  than  TT,  both 
L  and  M  must  be  zero  and  the  system  cyclic.  We  combine  this 
result  with  the  preceding  theorem  to  obtain  the  following: 

A  two-parameter  family  of  circles  through  a  point  and  orthogonal 
to  any  surface  constitute  a  cyclic  system,  and  the  most  general  spher 
ical  representation  of  the  developables  of  a  cyclic  congruence  is  afforded 
by  the  representation  of  the  axes  of  such  a  system  of  circles.^ 

*  Vol.  II,  p.  169.  t  Bianchi,  Vol.  II,  p.  170. 


442  CYCLIC  SYSTEMS 

We  consider  finally  the  case  where  the  planes  of  the  circles 
depend  upon  a  single  parameter.  If  we  take  for  moving  axes  the 
tangent,  principal  normal,  and  binormal  of  the  edge  of  regression 
of  these  planes  and  its  arc  for  the  parameter  w,  we  have 


and  comparing  (V,  50)  with  (2),  we  see  that 

p  —  --  »         <7  =  0,         r  =  -» 

T  p 

where  p  and  r  are  the  radii  of  first  and  second  curvature  of  the 
edge  of  regression.  Now 

da  ,«      b  ,       8a  „_#,«  r>       *b 

A  =  --  hi  --  »         A=  —  »         B  =  --  h-»        2?!  =  —  • 

du  p  dv  du      p  dv 

The  equations  (12)  reduce  to  two.  One  of  the  functions  a,  b  may 
be  chosen  arbitrarily  ;  then  the  other  and  R  can  be  obtained  by 
the  solution  of  partial  differential  equations  of  the  first  order. 


EXAMPLES 

1.  Show  that  a  congruence  of  Ribaucour  whose  surface  generator  is  the  right 
helicoid  is  cyclic,  and  determine  the  cyclic  systems. 

2.  A  congruence  of  Guichard  is  a  cyclic  congruence,  and  the  envelope  of  the 
planes  of  the  circles  of  each  associated  cyclic  system  is  a  surface  of  Voss. 

3.  The  surface  generator  of  a  cyclic  congruence  of  Ribaucour  is  an  associate 
surface  of  the  planes  of  the  circles  of  each  associated  cyclic  system. 

4.  If  S  is  a  surface  whose  lines  of  curvature  have  the  same  spherical  representa 
tion  as  a  pseudospherical  surface,  and  Si  is  a  transform  of  S  resulting  from  a  gen 
eralized  Bianchi  transformation  (§  178),  the  tangents  to  the  lines  of  curvature  of 
81  pass  through  the  centers  of  principal  curvature  of  S. 

5.  When  the  focal  segment  of  each  line  of  a  cyclic  congruence  is  divided  in 
constant  ratio  by  the  center  of  the  circle,  the  envelope  of  the  planes  of  the  circles 
is  a  surface  of  Voss. 

6.  The  circles  of  the  cyclic  system  whose  axes  are  normal  to  the  surface  S, 
defined  in  Ex.  11,  p.  370,  pass  through  a  point,  and  the  surfaces  orthogonal  to  the 
circles  are  surfaces  of  Bianchi  of  the  parabolic  type. 

7.  If  the  spheres  with  the  focal  segments  of  the  lines  of  a  congruence  for 
diameters  pass  through  a  point,  the  congruence  is  cyclic,  and  the  circles  pass 
through  the  point. 

8.  Show  that  the  converse  of  Ex.  7  is  true. 


GENERAL  EXAMPLES  443 

GENERAL  EXAMPLES 

1.  Determine  the  normal  congruences  of  Ribaucour  which  are  cyclic. 

2.  If  the  envelope  of  the  planes  of  the  circles  of  a  cyclic  system  is  a  surface  of 
Voss  whose  conjugate  geodesic  system  corresponds  to  the  developables  of  the  asso 
ciated  cyclic  congruence,  any  family  of  planes  cutting  the  focal  segments  in  con 
stant  ratio  and  perpendicular  to  them  envelop  a  surface  of  Voss. 

3.  A  necessary  and  sufficient  condition  that  a  congruence  be  cyclic  is  that  the 
developables  have  the  same  spherical  representation  as  the  conjugate  lines  of  a  sur 
face  which  remain  conjugate  in  a  deformation  of  the  surface.    If  the  developables 
of  the  congruence  are  real,  the  deforms  of  the  surface  are  imaginary. 

4.  The  planes  of  the  cyclic  systems  associated  with  a  cyclic  congruence  of 
Ribaucour  touch  their  respective  envelopes  in  such  a  way  that  the  points  of  con 
tact  of  all  the  planes  corresponding  to  the  same  line  of  the  congruence  lie  on  a 
straight  line. 

5.  If  the  spheres  described  on  the  focal  segments  of  a  congruence  as  diameters 
cut  a  fixed  sphere  orthogonally  or  in  great  circles,  the  congruence  is  cyclic  and 
the  circles  cut  the  fixed  sphere  orthogonally  or  in  diametrically  opposite  points. 

6.  If  one  draws  the  circles  which  are  normal  to  a  surface  S  and  which  cut 
a  fixed  sphere  SQ  in  diametrically  opposite  points  or  orthogonally,  the  spheres 
described  on  the  focal  segments  of  the  congruence  of  axes  as  diameters  cut  SQ  in 
great  circles  or  orthogonally. 

7.  Determine  the  cyclic  systems  of  equal  circles  whose  planes  envelop  a  devel 
opable  surface. 

8.  Let  Si  be  the  surface  defined  in  Ex.  14,  p.  371,  and  let  S0  be  the  sphere  with 
center  at  the  origin  and  radius  r.    Draw  the  circles  which  are  normal  to  Si  and 
which  cut   S0  orthogonally  or  in  diametrically  opposite  points.     Show  that  the 
cyclic  congruence  of  the  axes  of  these  circles  is  a  normal  congruence,  and  that  the 
coordinates  of  the  normal  surfaces  are  of  the  form 

[1  (      --  -)  T 

—  j  a?e    a  -  (r?2  +  K)  e«    cos  6  -f  77  sin  8  \Xi 

+  fJL  ja2e~«  -  (T;2  +  K)(P  I  sind  -  rj  cose]  X2  +  tX, 
L2«  (  ) 

where  K  is  equal  to  —  r2  or  +  r2,  according  as  the  circles  cut  S0  orthogonally  or  in 
diametrically  opposite  points,  and  where  t  is  given  by 


[1    (      -£  ^)  "1 

—  \a2e   a  -  (i?2  -I-  K)  ea  [  cos  6  +  -r\  sin  6   sin  «  du 
2a( 

[1    (      --  -)  ~\ 

_  j  a?e   a  —  (»j2  4-  K)  e"  j  sin  6  —  t\  cos  0   cos  u  dv. 


9.  Show  that  the  surfaces  of  Ex.  8  are  surfaces  of  Bianchi  which  have  the 
same  spherical  representation  of  their  lines  of  curvature  as  the  pseudospherical 
surface  S  referred  to  in  Ex.  14,  p.  371. 

10.  Show  that  the  surfaces  orthogonal  to  the  cyclic  system  of  Ex.  8  are  surfaces 
of  Bianchi  of  the  parabolic  type. 


444  CYCLIC  SYSTEMS 

11.  Let  S  be  a  surface  referred  to  an  orthogonal  system,  and  let  T  be  the  trihe 
dral  whose  x-axis  is  tangent  to  the  curve  u  =  const.    The  equations  . 

x  =  p(l  +  cos0),        y  =  0,        z  =  />sin0 

define  a  circle  normal  to  S.    Show  that  the  necessary  and  sufficient  conditions  that 
the  circles  so  defined  form  a  cyclic  system  are 


cu 

12.  A  necessary  and  sufficient  condition  that  a  cyclic  system  remain  cyclic 
when  an  orthogonal  surface  S  is  deformed  is  that  S  be  applicable  to  a  surface  of 
revolution  and  that 


where  c  is  a  constant  and  the  linear  element  of  S  is  ds2  =  du2  +  02(w)  dv*  (cf .  Ex.  11). 

13.  Determine  under  what  conditions  the  lines  of  intersection  of  the  planes  of 
the  circles  of  a  cyclic  system  and  the  tangent  planes  to  an  orthogonal  surface  form 
a  normal  congruence. 

14.  Let  Si  and  S2  be  two  surfaces  orthogonal  to  a  cyclic  system,  and  let  MI  and 
M2  be  the  points  of  intersection  of  one  of  the  circles  with  Si  and  S2.    Show  that 
the  normals  to  Si  and  S2  at  the  points  MI  and  M2  meet  in  a  point  M  equidistant 
from  these  points,  and  show  that  Si  and  S2  constitute  the  sheets  of  the  envelope  of 
a  two-parameter  family  of  spheres  such  that  the  lines  of  curvature  on  Si  and  S2 
correspond. 

15.  Let  -S  be  the  surface  of  centers  of  a  two-parameter  family  of  spheres  of 
variable  radius  JR,  and  let  Si  and  S2  denote  the  two  sheets  of  the  envelope  of  these 
spheres.    Show  that  the  points  of  contact  MI  and  M2  of  a  sphere  with  these  sheets 
are  symmetric  with  respect  to  the  tangent  plane  to  S  at  the  corresponding  point  M. 
Let  S  be  referred  to  a  moving  trihedral  whose  plane  y  =  0  is  the  plane  MiMM2,  and 
let  the  parametric  curves  be  tangent  to  the  x-  and  y-axes  respectively.    Show  that 
if  ff  denotes  the  angle  which  the  radius  MMi  makes  with  the  x-axis  of  the  trihedral, 
the  lines  of  curvature  on  Si  are  given  by 

£  sin  <r  (sin <rp  —  r  cos <r) du2  +  in[qi ) dv2 

\         dv/ 

•H  fllfl -)  —  I  sin  <r  (cos  <rri  +  p  sin  <r)  \dudv  =  0. 

16.  Find  the  condition  that  the  lines  of  curvature  on  S!  and  S2  of  Ex.  15  corre 
spond,  and  show  that  in  this  case  these  curves  correspond  to  a  conjugate  system  on  S. 

17.  Show  that  the  circles  orthogonal  to  two  surfaces  form  a  cyclic  system,  pro 
vided  that  the  lines  of  curvature  on  the  two  surfaces  correspond. 

18.  Let  <S  be  a  pseudospherical  surface  with  the  linear  element  (VIII,  22),  the 
lines  of  curvature  being  parametric,  and  let  A  be  a  surface  with  the  same  spher 
ical  representation  of  its  lines  of  curvature  as  S ;  furthermore,  let  AI  denote  the 
envelope  of  the  plane  which  makes  the  constant  angle  a  with  the  tangent  plane 
at  a  point  M  of  A  and  meets  this  plane  in  a  line  I/,  which  forms  with  the  tangent 
to  the  curve  u  =  const,  at  M  an  angle  0  defined  by  equations  (VIII,  35).   If  MI 


GENERAL  EXAMPLES  445 

denotes  the  point  of  contact  of  this  plane,  we  drop  from  MI  a  perpendicular  on  L, 
meeting  the  latter  in  N.  Show  that  if  X  and  p  denote  the  lengths  MN  and  NMi, 
they  are  given  by 

X  =  ( V2?  cos o>  +  V6?  sin o>) sin  <r,        ^  =  (  —  Vj£ sin  w  +  Vt?  cos  w) sin o-, 
where  E  and  (?  are  the  first  fundamental  coefficients  of  A. 

19.  Show  that  when  the  surface  A  in  Ex.  18  is  the  pseudospherical  surface  S, 
then  AI  is  the  Backhand  transform  Si  of  S  by  means  of  the  functions  (0,  <r),  and 
that  when  A  is  other  than  S  the  lines  of  curvature  on  the  four  surfaces  S,  .4,  Si, 
AI  correspond,  and  the  last  two  have  the  same  spherical  representation. 

20.  Show  that  as  0  is  given  all  values  satisfying  equations  (VIII,  35)  for  a  given 
0-,  the  locus  of  the  point  Jfi,  defined  in  Ex.  18,  is  a  circle  whose  axis  is  normal  to 
the  surface  A  at  M. 

21.  Show  that  when  A  in  Ex.  18  is  a  surface  of  Bianchi  of  the  parabolic  type 
(Ex.  11,  p.  370)  the  surfaces  AI  are  of  the  same  kind,  whatever  be  a-. 


CHAPTER  XIV 

TRIPLY  ORTHOGONAL  SYSTEMS  OF  SURFACES 

181.  Triple  system  of  surfaces  associated  with  a  cyclic  system. 
Let  S1  be  one  of  the  surfaces  orthogonal  to  a  cyclic  system,  and 
let  its  lines  of  curvature  be  parametric.  The  locus  2t  of  the 
circles  which  meet  Sl  in  the  line  of  curvature  v  =  const,  through 
a  point  M  is  a  surface  which  cuts  Sl  orthogonally.  Hence,  by 
Joachimsthal's  theorem  (§  59),  the  line  of  intersection  is  a  line  of 
curvature  for  2r  In  like  manner,  the  locus  22  of  the  circles 
which  meet  S:  in  the  line  of  curvature  u  =  const,  through  M  cuts 
S^  orthogonally,  and  the  curve  of  intersection  is  a  line  of  curva 
ture  on  S2  also.  Since  the  developables  of  the  associated  cyclic 
congruence  correspond  to  the  lines  of  curvature  on  all  of  the 
orthogonal  surfaces,  each  of  the  latter  is  met  by  2X  and  22  in  a 
line  of  curvature  of  both  surfaces.  At  each  point  of  the  circle 
through  M  the  tangent  to  the  circle  is  perpendicular  to  the  line 
of  curvature  v  =  const,  on  2t  through  the  point  and  to  u  =  const, 
on  2a.  Hence  the  circle  is  a  line  of  curvature  for  both  2X  and  22, 
and  these  surfaces  cut  one  another  orthogonally  along  the  circle. 

Since  there  is  a  surface  2X  for  each  curve  v  =  const,  on  Sl  and  a 
surface  22  for  each  u  =  const.,  the  circles  of  a  cyclic  system  and 
the  orthogonal  surfaces  may  be  looked  upon  as  a  system  of  three 
families  of  surfaces  such  that  through  each  point  in  space  there 
passes  a  surface  of  each  family.  Moreover,  each  of  these  three  sur 
faces  meets  the  other  two  orthogonally,  and  each  curve  of  intersec 
tion  is  a  line  of  curvature  on  both  surfaces.  We  have  seen  (§  96) 
that  the  confocal  quadrics  form  such  a  system  of  surfaces,  and 
another  example  is  afforded  by  a  family  of  parallel  surfaces  and 
the  developables  of  the  congruence  of  normals  to  these  surfaces. 

When  three  families  of  surfaces  are  so  constituted  that  through 
each  point  of  space  there  passes  a  surface  of  each  family  and  each 
of  the  three  surfaces  meets  the  other  two  orthogonally,  they  are 

446 


GENERAL  EQUATIONS  447 

said  to  form  a  triply  orthogonal  system.  In  the  preceding  examples 
the  curve  of  intersection  of  any  two  surfaces  is  a  line  of  curvature 
for  both.  Dupin  showed  that  this  is  a  property  of  all  triply  orthog 
onal  systems.  We  shall  prove  this  theorem  in  the  next  section. 

182.  General  equations.  Theorem  of  Dupin.  The  simplest  exam 
ple  of  an  orthogonal  system  is  afforded  by  the  planes  parallel  to 
the  coordinate  planes.  The  equations  of  the  system  are 

3  =  1*!,         y  =  Ma,         z  =  i*8, 

where  u#  u^  u3  are  parameters.  Evidently  the  values  of  these 
parameters  corresponding  to  the  planes  through  a  point  are  the 
rectangular  coordinates  of  the  point.  In  like  manner,  the  surfaces 
of  each  family  of  any  triply  orthogonal  system  may  be  determined 
by  a  parameter,  and  the  values  of  the  three  parameters  for  the 
three  surfaces  through  a  point  constitute  the  curvilinear  coordi 
nates  of  the  point.  Between  the  latter  and  the  rectangular  coor 
dinates  there  obtain  equations  of  the  form 
(1)  x  =/1(w1,  i*8,  i*,),  y  =/2K,  i*a,  i*,),  z  =/,(!*!,  i*a,  i*8), 
where  the  functions  /  are  analytic  in  the  domain  considered.  An 
example  of  this  is  afforded  by  formulas  (VII,  8),  which  define  space 
referred  to  a  system  of  confocal  quadrics. 

In  order  that  the  system  be  orthogonal  it  is  necessary  and  suffi 
cient  that  these  functions  satisfy  the  three  conditions 

v  dx   dx  _  v  dx  dx  y  dx  dx  _ 

^aST          *totto.-          Zto.dut 

Any  one  of  the  surfaces  ut=  const,  is  defined  by  (1)  when  ut  is 
given  this  constant  value. 

By  the  linear  element  of  space  at  a  point  we  mean  the  linear  ele 
ment  at  the  point  of  any  curve  through  it.  This  is 


which,  in  consequence  of  (2),  may  be  written  in  the  parametric  form 
(3)  ds2  =  H'l  du*  +  H*  du*  +  HI  dui, 


As  thus  defined,  the  functions  H#  H2,  H3  are  real  and  we  shall 
assume  that  they  are  positive. 


448     TRIPLY  ORTHOGONAL  SYSTEMS  OF  SURFACES 


From  (3)  we  have  at  once  the  linear  element  of  any  of  the  sur 
faces  of  the  system.    For  instance,  the  linear  element  of  a  surface 

U=  COnst.   is  ,2         rrl  J. 


Now  we  shall  find  that  the  second  quadratic  forms  of  these  surfaces 
are  expressible  in  terms  of  the  functions  H  and  their  derivatives. 

If  X^  F.,  Zi  denote  the  direction-cosines  of  the  normals  to  the 
surfaces  ui  =  const,  we  have 


(5) 


,  du. 


We  choose  the  axes  such  that 


(6) 


=  4-1. 


In  consequence  of  (5)  the  second  fundamental  coefficients  of  a  sur 
face  u{  —  const,  are  defined  by 

dx  d*x         _,       1  ^  dx     d2x_  „  _  1  y  dx  d*x 

^ut  Hi  **  dut  du? 


_  1 
~ 


,=    1   y  dx 
4 


where  t,  /c,  I  take  the  values  1,  2,  3  in  cyclic  order,  and  the  sign  2 
refers  to  the  summation  of  terms  in  #,  ?/,  2,  as  formerly.  In  order 
to  evaluate  these  expressions  we  differentiate  equations  (2)  with 
respect  to  u^  uv  u2  respectively.  This  gives 

dx 


dx 


±±.  _JL±__  =  o,f 


dx 


^  du. 


0. 


If  each  of  these  equations  be  subtracted  from  one  half  of  the  sum 
of  the  three,  we  have 


=  o, 


dx     d*x 


^  duz  du 
consequently  D-=  0. 


du3  dul 


=  0,     V 


=  0; 


THEOREM  OF  DUPIN.     EQUATIONS  OF  LAME       449 

If  the  first  and  third  of  (2)  be  differentiated  with  respect  to 
uz  and  us  respectively,  and  the  second  and  third  of  (4)  with 
respect  to  uv  we  have 

dx  d*z  dx      tfx  a#2 

2  ~~         2  du^  ' 

dx  tfx          y  dx      tfx  3HS 

~          3         ' 


J5Tt 


Hence  we  have 


Proceeding  in  like  manner,  we  find  the  expressions  for  the  other 
Z>'s,  which  we  write  as  follows  : 


where  i,  K,  I  take  the  values  1,  2,  3  in  cyclic  order.  From  the  sec 
ond  of  these  equations  and  the  fact  that  the  parametric  system  on 
each  surface  is  orthogonal,  follows  the  theorem  of  Dupin  : 

The  surfaces  of  a  triply  orthogonal  system  meet  one  another  in 
lines  of  curvature  of  each. 

183.  Equations  of  Lame".  By  means  of  these  results  we  find  the 
conditions  to  be  satisfied  by  Hv  H^  Hz,  in  order  that  (3)  may  be 
the  linear  element  of  space  referred  to  a  triply  orthogonal  system 
of  surfaces.  For  each  surface  the  Codazzi  and  Gauss  equations 
must  be  satisfied.  When  the  above  values  are  substituted  in  these 
equations,  we  find  the  following  six  equations  which  it  is  necessary 
and  sufficient  that  the  functions  H  satisfy  : 

PH{         1    dH^H,       1  gJgjgjr^ 

~' 


/ox 

H 

where  t,  /^,  I  take  the  values  1,  2,  3  in  cyclic  order.  These  are 
the  equations  of  Lame",  being  named  for  the  geometer  who  first 
deduced  them.* 

*  Lemons  sur  les  coordonntes  curvilignes  et  leurs  diverses  applications,  pp.  73-79. 
Paris,  1859. 


450  TRIPLY  ORTHOGONAL  SYSTEMS  OF  SURFACES 

For  each  of  the  surfaces  there  is  a  system  of  equations  of  the 
form  (V,  16).  When  the  values  from  (7)  are  substituted  in  these 
equations  we  have 


—  < 

t  du, 


Recalling  the  results  of  §  65,  we  have  that  each  set  of  solutions 
of  equations  (8),  (9)  determine  a  triply  orthogonal  system,  unique 
to  within  a  motion  in  space.  In  order  to  obtain  the  coordinates 
of  space  referred'  to  this  system,  we  must  find  nine  functions 
JQ,  r;.,  Zf  which  satisfy  (10)  and 


1,         2^=0.  (»=*=*) 

Then  the  coordinates  of  space  are  given  by  quadratures  of  the  form 
x  =  I  H^XI  dul  +  H^XZ  du2  +  HZXZ  dus  . 

If  p.K  denotes  the  principal  radius  of  a  surface  uf=  const,  in  the 
direction  of  the  curve  of  parameter  UK,  we  have,  from  (7), 

rm  1- 

Pi. 


Let  pl  denote  the  radius  of  first  curvature  of  a  curve  of  param 
eter  ur  In  accordance  with  §  49  we  let  w1  and  w[—  ?r/2  denote  the 
angles  which  the  tangents  to  the  curves  of  parameter  u3  and  u2 
respectively  through  the  given  point  make,  in  the  positive  sense, 
with  the  positive  direction  of  the  principal  normal  of  the  curve 
of  parameter  ur  Hence,  by  (IV,  16),  we  have 

/i  o\ 

Pi  Pn  Pi  Pzi 

From  these  equations  and  similar  ones  for  curves  of  parameter  u 
and  us,  we  deduce  the  relations 

(13)  1  =  1  +  1,  tan5(  =  6t, 

Pi      Pl>     Pfi  P« 


ONE  FAMILY  OP  SURFACES  OF  REVOLUTION      451 

where  2,  /c,  I  take  the  values  1,  2,  3  in  cyclic  order.  Moreover,  since 
the  parametric  curves  are  lines  of  curvature,  it  follows  from  (§59) 
that  the  torsion  of  a  curve  of  parameter  ui  is 

(14)  l-l^i. 

r{      Hi  du, 

184.  Triple  systems  containing  one  family  of  surfaces  of  revolution. 

Given  a  family  of  plane  curves  and  their  orthogonal  trajectories  ; 
if  the  plane  be  revolved  about  a  line  of  the  plane  as  an  axis,  the 
two  families  of  surfaces  of  revolution  thus  generated,  and  the  planes 
through  the  axis,  form  a  triply  orthogonal  system.  We  inquire 
whether  there  are  any  other  triple  systems  containing  a  family  of 
surfaces  of  revolution. 

Suppose  that  the  surfaces  us  =  const,  of  a  triple  system  are  sur 
faces  of  revolution,  and  that  the  curves  u2  =  const,  upon  them  are 
the  meridians.  Since  the  latter  are  geodesies,  we  must  have 


From  (8)  it  follows  that  either 

dH.      n  dffs      n 

—  i  =  0,      or      —  8  =  0. 

dus  du2 

In  the  first  case  it  follows  from  (11)  that  l//o31  =  0.  Consequently, 
the  surfaces  of  revolution  w3  =  const,  are  developables,  that  is,  either 
circular  cylinders  or  circular  cones.  Furthermore,  from  (15)  and 
(11),  we  have  l//o21=0,  so  that  the  surfaces  u2  =  const,  also  are 
developables,  and  in  addition  we  have,  from  (13),  that  l//^  =  0,  that 
is,  the  curves  of  parameter  u^  are  straight  lines  and  consequently 
the  surfaces  u^=  const,  are  parallel.  The  latter  are  planes  when 
the  surfaces  us  =  const,  are  cylinders,  and  surfaces  with  circular 
lines  of  curvature  when  us  =  const,  are  circular  cones.  Conversely, 
from  the  theorem  of  Darboux  (§  187)  and  from  §  132,  it  follows 
that  any  system  of  circular  cylinders  with  parallel  generators,  or 
any  family  of  circular  cones  whose  axes  are  tangent  to  the  locus 
of  the  vertex,  leads  to  a  triple  system  of  the  kind  sought. 

We  consider  now  the  second  case,  namely 


452     TRIPLY  ORTHOGONAL  SYSTEMS  OF  SURFACES 

From  (11)  we  find  that  l//o21=0,  and  l//o28  =  0;  consequently  the 
surfaces  wa=  const,  are  planes.  Since  these  are  the  planes  of  the 
meridians,  it  follows  that  the  axes  of  the  surfaces  coincide,  and 
consequently  the  case  cited  at  the  beginning  of  this  section  is  the 
only  one  for  nondevelopable  surfaces. 

185.  Triple  systems  of  Bianchi  and  of  Weingarten.  In  §  119 
it  was  found  that  all  the  Bianchi  transforms  of  a  given  pseudo- 
spherical  surface  are  pseudospherical  surfaces  of  the  same  total 
curvature,  and  that  they  are  the  orthogonal  surfaces  of  a  cyclic 
system  of  circles  of  constant  radius.  Hence  the  totality  of  these 
circles  and  surfaces  constitutes  a  triply  orthogonal  system,  such 
that  the  surfaces  in  one  family  are  pseudospherical.  As  systems 
of  this  sort  were  first  considered  by  Ribaucour  (cf.  §  119),  they 
are  called  the  triple  si/stems  of  Ribaucour.  We  proceed  to  the 
consideration  of  all  triple  systems  such  that  the  surfaces  of  one 
family  are  pseudospherical.  These  systems  were  first  studied  by 
Bianchi,  *  and  consequently  Darboux  f  has  called  them  the  systems 
of  Bianchi. 

From  §  119  it  follows  that  the  parameters  of  the  lines  of  curva 
ture  of  a  pseudospherical  surface  of  curvature  —  l/aa  can  be  so 
chosen  that  the  linear  element  takes  the  form 

(1 6)  d «a  =  cosa  o)  du*  +  sina  o>  dv\ 

where  o>  is  a  solution  of  the  equation 

dao>      dao>  _  sin  o>  cos  a) 

In  this  case  the  principal  radii  are  given  by 

1  tan  &)  1       cot  a) 

(18) 

Pl  a  p,         a 

In  general  the  total  curvature  of  the  pseudospherical  surfaces 
of  a  system  of  Bianchi  varies  with  the  surfaces.  If  the  surfaces 
w3  =  const,  are  the  pseudospherical  surfaces,  we  may  write  the 
curvature  in  the  form  —  1/f^,  where  ?/8  is  a  function  of  ?/„  alone. 

•Annali,  8er.  2,  Vol.  XIII  (188,",),  pp.  177-234;  Vol.  XIV  (1880),  pp.  115-130;  Lezioni, 
Vol.  II,  chap,  xxvii. 

t  7>vo»w  «wr  les  ni/stemes  orthogonaux  et  les  coonlonntes  curvilignes,  pp.  308-323. 
Paris,  18U8. 


(19) 


TRIFLE  SYSTEMS  OF  BIANCHI  453 

In  accordance  with  (11)  and  (18)  we  put 
_1  1      dff,      tan<w 

P* 


, 


1      g//  cot  <a 


s)  tf  ?)  ff 

If  these  values  of  - — -  and  — =  be  substituted  in  equations  (8)  for* 

r/r,  (K,, 

equal  to  1  and  2  respectively,  we  obtain 

1   BJf,  3to  1    dlfa  do) 

l  =  —  tan  o>  — , =  =  cot  a) 

From  these  equations  we  have,  by  integration, 

(20)  ffl  =  <£13  •  cos  co,          Hz  =  </>.,3  •  sin  CD, 

where  $13  and  $23  are  functions  independent  of  wa  and  uv  respectively. 

\Vo  shall  show  that  both  of  them  are  independent  of  u3. 

When  the  values  of  Hl  and  !!„  from  (20)  are  substituted  in  (19), 
we  have  respectively 


(21) 


/.          to      3  log  <£. 
jff  =  f  r  Cot  cw  (  tan  to—  I 


/  da)       d  log  A0  \ 

//  =  f  r  tan  o>  [  cot  o>  ---  \-  -      •z»  ! 

\         ,5w,          ^3     / 
From  these  equations  it  follows  that 


Hence,  unless  0l;l  and  ^>.,;l  are  independent  of  w3,  tan  o>  is  equal  to 
the  ratio  of  a  function  of  uv  and  w3  and  of  a  function  of  w2  and  ?<3. 
We  consider  the  latter  case  and  study  for  the  moment  a  partic 
ular  surface  i/3=  c.    By  the  change  of  parameters 


(^..(MP  cjau^ 
the  linear  element  of  the  surface  reduces  to  (16),  and  (22)  becomes 

tan  co  =  — » 

where  f  and  V  are  functions  of  u  and  v  respectively.    When  this 
is  substituted  in  (17),  we  obtain 

a  ,    I--' 

/C\^-  i     *S  '  \     *  -w-r-ft       .        w^Ox  V 


,,,       ivn  ,  v 

(u"T 


454     TKIPLY  ORTHOGONAL  SYSTEMS  OF  SURFACES 

If  this  equation  be  differentiated  successively  with  respect  to  u 
and  v,  we  find         /U"\f  1         /V"\r   1 

\u)  ~uu~'  +  \r)  ~vv'  =  °» 

unless  V  or  V'  is  equal  to  zero.    From  this  it  follows  that 


where  K  denotes  a  constant.    Integrating,  we  have 
U"=2icU*+aU,  V"=-2tcVs  +  l 

a  and  y3  being  constants,  and  another  integration  gives 

U'*=KU*+a(r*+<y,         F'2=-*F4  +  /3F2+S. 
When  these  expressions  are  substituted  in  (23),  we  find 


This  condition  can  be  satisfied  only  when  the  curvature  is  zero. 
Hence  U1  or  V'  must  be  zero,  that  is,  &)  must  be  a  function  of  u  or  v 
alone.  In  this  case  the  surface  is  a  surface  of  revolution.  In  accord 
ance  with  §  184  a  triple  system  of  Bianchi  arises  from  an  infinity 
of  pseudospherical  surfaces  of  revolution  with  the  same  axis. 

When  exception  is  made  of  this  case,  the  functions  (f>13  and  </>23 
in  (20)  are  independent  of  us.    Hence  the  parameters  of  the  sys 
tems  may  be  chosen  so  that  we  have 
/2^x  H  —  IT—''  H—  f7 — 

When  these  values  are  substituted  in  the  six  equations  (8),  (9), 
they  reduce  to  the  four  equations 

~2&)      £2ft)      sin  to  cos  &) 

=  U, 


(25) 


duf      0M*  III 

cot  ft)  - —  — — —  -f-  tan  to  — =  0, 

0/1        g2ft)   \       1     d  /sinftA 1      g<»     £2ft)    =() 

cuv  \cos  ft)  du1duj      U3  du3\  U3  /      sin  G)  du2  du2du3 


_d_  /    1        d2&)    \       1     d   /cos  ft)\          1      do     d^co     _  Q 
du2  \sin  &)  du2  duj      U3  du3  \   U3   /      cos  ft)  cuv  dul  duz 


TRIPLE  SYSTEMS  OP  WEINGAKTEN  455 

Darboux  has  inquired  into  the  generality  of  the  solution  of  this 
system  of  equations,  and  he  has  found  that  the  general  solution 
involves  five  arbitrary  functions  of  a  single  variable.  We  shall 
not  give  a  proof  of  this  fact,  but  refer  the  reader  to  the  investi 
gation  of  Darboux.* 

We  turn  to  the  consideration  of  the  particular  case  where  the 
total  curvature  of  all  the  pseudospherical  surfaces  is  the  same, 
which  may  be  taken  to  be  —1  without  any  loss  of  generality. 
As  triple  systems  of  this  sort  were  first  discussed  by  Weingarten, 
we  follow  Bianchi  in  calling  them  systems  of  Weingarten.  Of  this 
kind  are  the  triple  systems  of  Ribaucour. 

For  this  case  we  have  U3  =  1,  so  that  the  linear  element  of  space  is 

(26)  ds2  =  cos2 &)  dul  +  sin2 a)  du* 

Since  the  second  of  equations  (25)  may  be  written  in  either  of 

the  forms         c   a    /    \        ffw    \  \      cw     cza) 

)= , 

n  ft)  du2  dus/         cos  ft)  du2  du^  dus 


du2  \cos  ft)  du^  du3/     sin  ft)  dul  du2  dus 

if   We   pUt  /       -j  2          \2        /      -j  o2          \2        /o 

ycos  ft)  du1  du^l     V  sin  &)  duz  cuzj      \^3 
it  follows  from  the  last  two  of  (25)  and  from  (27)  that 


Hence  <J>  is  a  function  of  us  alone.  But  by  changing  the  param 
eter  uz,  an  operation  which  will  not  affect  the  form  of  (26),  we  can 
give  <&  a  constant  value,  say  c.  Consequently  we  have 


(28)  --  -+  ---=  c. 

\cos  ft)  Buldu3J     \sm  co  du2du3/      \^3/ 

Bianchi  has  shown  f  that  equation  (28)  and  the  first  of  (25)  are 
equivalent  to  the  system  (25),  when  Z78  =  l.  Consequently  the 
problem  of  the  determination  of  triple  systems  of  Weingarten  is 
the  problem  of  finding  common  solutions  of  these  two  equations. 

*L.c.,  pp.  313,  314;  Bianchi,  Vol.  II,  pp.  531,  532.  t  Vol.  II,  p.  550. 


456     TRIPLY  ORTHOGONAL  SYSTEMS  OF  SURFACES 

EXAMPLES 

1.  Show  that  the  equations 

x  =  r  cos  u  cos  v,        y  =  r  cos  u  sin  u,        z  =:  r  sin  w 
define  space  referred  to  a  triply  orthogonal  system. 

2.  A  necessary  and  sufficient  condition  that  the  surfaces  us  =  const,  of  a  triply 
orthogonal  system  be  parallel  is  that  Hs  be  a  function  of  u3  alone.    What  are  the 
other  surfaces  u\  =  const.,  u%  =  const.? 

3.  Two  near-by  surfaces  us  =  const,  intercept  equal  segments  on  those  orthog 
onal  trajectories  of  the  surfaces  w3  —  const,  which  pass  through  a  curve  Hs  =  const. 
on  the  former;  on  this  account  the  curves  #3  =  const,  on  the  surfaces  u3  =  const. 
are  called  curves  of  equidistance. 

4.  Let  the  surfaces  w3  =  const,  of  a  triple  system  be  different  positions  of  the 
same  pseudosphere,  obtained  by  translating  the  surface  in  the  direction  of  its  axis. 
Determine  the  character  of  the  other  surfaces  of  the  system. 

5.  Derive  the  following  results  for  a  triple  system  of  Weingarten  : 


V/8w\2 
C+U 


where  the  differential  parameter  is  formed  with  respect  to  the  linear  element  of  a 

surface  u3  =  const.,  and  pg  is  the  radius  of  geodesic  curvature  of  a  curve  —  =  const 

£w3 

on  this  surface.   Show  that  the  curves  of  equidistance  on  the  surfaces  us  =  const. 
are  geodesic  parallels  of  constant  geodesic  curvature. 

6.  Show  that  when  c  in  (28)  is  equal  to  zero,  the  first  curvature  l/p8  of  the 
curves  of  parameter  u3  is  constant  and  equal  to  unity;  that  equations  similar 
to  (12)  become 

2  _    8<a  a2u>  dot 

=  —  sin  o>  cos  w3  —  ,  —  =  —  cos  u»  sin  u>3  —  ; 


that  if  we  put  0  —  -  —  w3,  the  last  two  of  equations  (25),  where  U&  =  1,  may  be 
written 


cd       gw  dO       aw 

--  1  --  =  sm  6  cos  w,         —  ^  H  --  =  —  cos  0  sin  u  ; 


&d  /    1         8*6    \2     /    1         8*6    \2     /de\* 

--  ^  =  Sin  6  COS  0,  /  ---  J   -f  /  ---  ]   =  /  -  )  . 

8u  cos6 


and  that 


When  c  =  0  in  (28)  the  system  is  said  to  be  of  constant  curvature. 

7.  A  necessary  and  sufficient  condition  that  the  curves  of  parameter  us  of  a 
system  of  Weingarten  be  circles  is  that  w3  be  independent  of  u3.  In  this  case 
(cf.  Ex.  6)  the  surfaces  us  =  const,  are  the  Bianchi  transforms  of  the  pseudo- 
spherical  surface  with  the  linear  element 

ds*  =  cos*0du* 


THEOREM  OF  RIBAUCOUR  457 

186.  Theorem  of  Ribaucour.  The  following  theorem  is  due  to 
Ribaucour  *  : 

Griven  a  family  of  surfaces  of  a  triply  orthogonal  system  and  their 
orthogonal  trajectories;  the  osculating  circles  to  the  latter  at  their 
points  of  meeting  with  any  surface  of  the  family  form  a  cyclic  system. 

In  proving  this  theorem  we  first  derive  the  conditions  to  be  satis 
fied  by  a  system  of  circles  orthogonal  to  a  surface  S  so  that  they  may 
form  a  cyclic  system.  Let  the  lines  of  curvature  on  S  be  parametric 
and  refer  the  surface  to  the  moving  trihedral  whose  x-  and  ?/-axes 
are  tangent  to  the  curves  v  =  const.,  u  =  const.  We  have  (V,  63) 

(29)  ^=n=p  =  qi=0. 

If  <£  denotes  the  angle  which  the  plane  of  the  circle  through  a 
point  makes  with  the  corresponding  zz-plane,  0  the  angle  which 
the  radius  to  a  point  P  of  the  circle  makes  with  its  projection  in 
the  z^-plane,  and  R  the  radius  of  the  circle,  the  coordinates  of  P 
with  reference  to  the  moving  axes  are 

x  =  R(\  -f  cos  0)  cos  $,     y  =^(1+  cos  0)  sine/),     z=lism0. 

Moreover,  the  direction-cosines  of  the  tangent  to  the  circle  at  P  are 

—  sin  6  cos  <£,         —  sin  6  sin  $,         cos  6. 

If  we  express  the  condition  that  every  displacement  of  P  must  be 
at  right  angles  to  this  line,  we  have,  from  (29)  and  (V,  51), 


dB  -  [sin  B(  -  —  +  lL^i)+  q  cos  0(1  +  cos  0)1  du 
[_         \R  du          R     /  J 

f  .     J\  dR      77,  sin<f>\  -."I  7 

-   sin  6  -  —  +  A_  _¥  _  p  sm  0(i  +  cos  B)\dv  =  0. 
[_  \jri  cv  K      /  J 

The  condition  that  this  equation  admit  an  integral  is  reducible  to 

cosjAI  [sin  4,  cose/) 

E     /J  L        R 


Hence,  as  remarked  before  (§  174),  if  there  are  three  surfaces  orthog 
onal  to  a  system  of  circles,  the  system  is  cyclic. 

*  Comptes  Rendus,  Vol.  LXX  (1870),  pp.  330-333. 


458     TRIPLY  ORTHOGONAL  SYSTEMS  OF  SURFACES 
The  condition  that  it  be  cyclic  is 


(30) 


d_  /i?1  sin  (f>\      d_ 


cu  \     R      /      dv\     R 
sin  <f>  cos  (/ 


,        d  /sin  (f)     \      d  icos<f)   \_ /> 


Since  the  principal  radii  of  S  are  given  by 

(31)  i  =  -|.       i  =  J. 

the  second  of  equations  (30)  reduces  to  the  first  when  S  is  a  sphere 
or  a  plane.    Hence  we  have  incidentally  the  theorem : 

A  two-parameter  system  of  circles  orthogonal  to  a  sphere  and  to 
any  other  surface  constitute  a  cyclic  system. 

We  return  to  the  proof  of  the  theorem  of  Ribaucour  and  apply 
the  foregoing  results  to  the  system  of  osculating  circles  of  the 
curves  of  parameter  u3  of  an  orthogonal  system  at  their  points  of 
intersection  with  a  surface  u»=  const. 

o 

From  equations  similar  to  (12)  we  have,  by  (11), 

cos  <j)  1      d//.,  sin  <f>  1      dHz 

and  the  equations  analogous  to  (31)  are 

1  q  1      211^  1    _  pl  _          1      3HZ 


pn          ^          H^HZ  du3 />,2      //2          7/2//3  dus 

When  these  values  are  substituted  in  equations  (30)  the  first 
vanishes  identically,  likewise  the  second,  in  consequence  of  equa 
tions  (8).  Hence  the  theorem  of  Ribaucour  is  proved.* 

187.  Theorems  of  Darboux.  The  question  naturally  arises 
whether  any  family  of  surfaces  whatever  forms  part  of  a  triply 
orthogonal  system.  This  question  will  be  answered  with  the  aid 
of  the  following  theorem  of  Darboux, f  which  we  establish  by  his 
methods : 

A  necessary  and  sufficient  condition  that  two  families  of  surfaces 
orthogonal  to  one  another  admit  of  a  third  family  orthogonal  to  both 
is  that  the  first  two  meet  one  another  in  lines  of  curvature. 

*  For  a  geometrical  proof  the  reader  is  referred  to  Darboux,  I.e.,  p.  77.     t  L.c.,  pp.  6-8. 


THEOREM  OF  DARBOUX  459 

Let  the  two  families  of  surfaces  be  defined  by 
(32)  a(x,  y,  z)  =  a,         @(x,  y,  z)  =  b, 

where  a  and  b  are  the  parameters.    The  condition  of  orthogonality  is 

dx  ~dx      dy^y      ~dz~dz~ 

In  order  that  a  third  family  of  surfaces  exist  orthogonal  to  the 
surfaces  of  the  other  families,  there  must  be  a  function  7(2,  y,  z) 
satisfying  the  equations 


.  ,  _ 

dx  dx      dydy      dz  dz~ 


_ 

~dx'dx      ~d^       ~dz~dz~ 


If  dx,  dy,  dz  denote  the  projections  on  the  axes  of  a  displace 
ment  of  a  point  on  one  of  the  surfaces  7  =  const.,  we  must  have 

dx     dy     dz 

da      da      da 

dx      dy      dz  =  0. 

Idx     ~dy     ~dz 

This  equation  is  of  the  form  (XIII,  9).    The  condition  (XIII,  10) 
that  it  admit  of  an  integral  involving  a  parameter  is 

da 


dx  dz*      dz  dxdz      dx  dz*       dz  dxdz 

~^ydxdy~~dx^y*  +  'dydxdy'{'^x  ty*\~    ' 

where  S  indicates  the  sum  of  the  three  terms  obtained  by  permut 
ing  x,  y,  z  in  this  expression.    If  we  add  to  this  equation  the  identity 

d  (a,  /3)  \da  ^ 


01*  -^-\  v  f~f          u  f-f  ^-\  i/  LV  i          f* 

—  '  '  — 5 ^ —  '  '  — ~i  \  =~~     ' 

the  resulting  equation  may  be  written  in  the  form 


(34) 


da  dp      J     d/3\      ,/p   da 

T~  T~      °  a'^~'~  ^l&  — 

dx  dx        \      dx 

da  dj3 

da  d/3 

dz  dz 


S(      ill )_  gr/3  _ 

I          '        r\          I  I    *1        r\    . 


=  0, 


460  TRIPLY  ORTHOGONAL  SYSTEMS  OF  SURFACES 


where,  for  the  sake  of  brevity,   we  have  introduced  the  symbol 
8(0,  4>),  defined  by 


If  equation  (33)  be  differentiated  with  respect  to  x,  the  result  may 
be  written 


Consequently  equation  (34)  is  reducible  to 

da     df3 
dx      dx 

r\  „  O  /O 

ccc      cp 


(35) 


da 
dz 


dy 


dz 


=o, 


which  is  therefore  the  condition  upon  a  and  /S  in  order  that  the 
desired  function  7  exist. 

A  displacement  along  a  curve  orthogonal  to  the  surfaces  a = const, 
is  given  by  ^  =  ^_^. 

da      da      da 
dx      dy      dz 

Such  a  curve  lies  upon  a  surface  /3  =  const,  and  since,  by  (35), 
it  satisfies  the  condition 


=  0, 


it  is  a  line  of  curvature  on  the  surface  (cf.  Ex.  3,  p.  247).  Hence 
the  curves  of  intersection  of  the  surfaces  a  =  const.,  fi  =  const., 
being  the  orthogonal  trajectories  of  the  above  curves,  are  lines  of 
curvature  on  the  surfaces  ft  =  const.  And  by  Joachimsthal's  theo 
rem  (§  59)  they  are  lines  of  curvature  on  the  surfaces  a  =  const, 
also.  Having  thus  established  the  theorem  of  Darboux,  we  are  in 
a  position  to  answer  the  question  at  the  beginning  of  this  section. 


dx 
dy 
dz 

c/3 

dx 

a/8 
9y 

8/8 

dz 

0/3 

*5 
if 

dy 

a/3 

rffe 

TRANSFORMATION  OF  COMBESCURE  461 

Given  a  family  of  surfaces  a  —  const.  ;  the  lines  of  curvature  in 
one  family  form  a  congruence  of  curves  which  must  admit  a  family 
of  orthogonal  surfaces,  if  the  surfaces  a  =  const,  are  to  form  part 
of  an  orthogonal  system.  If  this  condition  is  satisfied,  then,  accord 
ing  to  the  theorem  of  Darboux,  there  is  a  third  family  of  surfaces 
which  together  with  the  other  two  form  an  orthogonal  system. 

If  Xv  Yv  Z1  denote  the  direction-cosines  of  the  tangents  to  the 
lines  of  curvature  in  one  family  on  the  surfaces  a  —  const.,  the  ana 
lytical  condition  that  there  be  a  family  of  surfaces  orthogonal  to 
these  curves  is  that  the  equation 


admit  an  integral  involving  a  parameter.    The  condition  for  this  is 


In  order  to  find  X^  Y^  Z1  we  remark  that  since  they  are  the  direc 
tion-cosines  of  the  tangents  to  a  line  of  curvature  we  must  have 


and  similar  equations  in  Y,  Z,  where  the  function  X  is  a  factor  of 
proportionality  to  be  determined  arid  Jf,  Y,  Z  are  the  direction- 
cosines  of  the  normal  to  the  surface  a  =  const.  Hence,  if  the 
surfaces  are  defined  by  a  =  const.,  the  functions  Xv  Y^  Z^  are 
expressible  in  terms  of  the  first  and  second  derivatives  of  a,  and 
so  equation  (36)  is  of  the  third  order  in  these  derivatives.  There 
fore  we  have  the  theorem  of  Darboux*: 

The  determination  of  all  triply  orthogonal  systems  requires  the 
integration  of  a  partial  differential  equation  of  the  third  order. 

Darboux  has  given  the  name  family  of  Lame  to  a  family  of 
surfaces  which  forms  part  of  a  triply  orthogonal  system. 

188.  Transformation  of  Combescure.  We  close  our  study  of  triply 
orthogonal  surfaces  with  an  exposition  of  the  transformation  of 
Combescure,^  by  means  of  which  from  a  given  orthogonal  system 
others  can  be  obtained  such  that  the  normals  to  the  surfaces  of 
one  system  are  parallel  to  the  normals  to  the  corresponding  sur 
faces  of  the  other  system  at  corresponding  points. 

*  L.c.,  p.  12.        f  Annales  de  I'Ecole  Normale  Superieure,  Vol.  IV  (1867),  pp.  102-122. 


462  TRIPLY  ORTHOGONAL  SYSTEMS  OF  SURFACES 

We  make  use  of  a  set  of  functions  /3iK,  introduced  by  Dar- 
boux  *  in  his  development  of  a  similar  transformation  in  space  of 
n  dimensions.  By  definition 


In  terms  of  these  functions  equations  (8),  (9)  are  expressible  in 
the  form 

<37>        $-«*    t+ 

and  formulas  (10)  become 

(38)  —  A.jr.-arr,, 


Equations  (37),  (38)  are  the  necessary  and  sufficient  conditions  that 
the  expression        ^  ^  +  ^  ^  +  ^  ^ 


be  an  exact  differential.    From  their  form  it  is  seen  that  if  we  have 
another  set  of  functions  H[,  772,  //8'  satisfying  the  six  conditions 

<39>  *---  ' 


where  the  functions  /3tK  have  the  same  values  as  for  the  given 
system,  the  expression 

XJI[  dUl  -f  JT2//2'  du2  +  Xfi  dua, 

and  similar  ones  in  F,  Z,  are  exact  differentials,  and  so  by  quadra 
tures  we  obtain  an  orthogonal  system  possessing  the  desired  property. 
In  order  to  ascertain  the  analytical  character  'of  this  problem, 
we  eliminate  H[  and  H^  from  equations  (39)  and  obtain  the  three 
equations  ' 


n 


_  .._  , 

-^  du,   cu^ 

The  general  integral  of  a  system  of  equations  of  this  kind  involves 
three  arbitrary  functions  each  of  a  single  parameter  ut.  When  one 

*L.c.,  p.  161. 


GENERAL  EXAMPLES  463 

has  an  integral,  the  corresponding  values  of  H^  H^  are  given  directly 
by  (39).    Hence  we  have  the  theorem  : 

With  every  triply  orthogonal  system  there  is  associated  an  infinity 
of  others,  depending  upon  three  arbitrary  functions,  such  that  the 
normals  to  the  surfaces  of  any  two  systems  at  corresponding  points 

are  parallel.* 

EXAMPLES 

1.  In  every  system  of  Weingarten  for  which  c  in  (28)  is  zero,  the  system  of  cir 
cles  osculating  the  curves  of  parameter  us  at  points  of  a  surface  w3  =  const,  form  a 
system  of  Ribaucour  (§  185). 

2.  If  the  orthogonal  trajectories  of  a  family  of  Lame"  are  twisted  curves  of 
the  same  constant  first  curvature,  the  surfaces  of  the  family  are  pseudospherical 
surfaces  of  equal  curvature. 

3.  Every  triply  orthogonal  system  which  is  derived  from  a  cyclic  system  by  a 
transformation  of  Combescure  possesses  one  family  of  plane  orthogonal  trajectories. 

4.  If  the  orthogonal  trajectories  of  a  family  of  Lame*  are  plane  curves,  the  cyclic 
system  of  circles  osculating  these  trajectories  at  the  points  of  any  surface  of  the 
family  may  be  obtained  from  the  given  system  by  a  transformation  of  Combescure. 

5.  Determine  the  triply  orthogonal  systems  which  result  from  the  application  of 
the  transformation  of  Combescure  to  a  system  of  Ribaucour  (§  185). 

GENERAL  EXAMPLES 

1.  If  an  inversion  by  reciprocal  radii  (§  80)  be  effected  upon  a  triply  orthogonal 
system,  the  resulting  system  will  be  of  the  same  kind. 

2.  Determine  the  character  of  the  surfaces  of  the  system  obtained  by  an  inversion 
from  the  system  of  Ex.  1,  §  185,  and  show  that  all  the  curves  of  intersection  are  circles. 

3.  Establish  the  existence  of  a  triply  orthogonal  system  of  spheres. 

4.  A  necessary  and  sufficient  condition  that  the  asymptotic  lines  correspond  on 
the  surfaces  u%  =  const,  of  a  triply  orthogonal  system  is  that  there  exist  a  relation 
of  the  form 


03  =    ? 

where  0j,  02»  <t>s  are  functions  independent  of  w3. 

5.  When  the  condition  of  Ex.  4  is  satisfied,  those  orthogonal  trajectories  of  the 
surfaces  us  =  const,  which  pass  through  points  of  an  asymptotic  line  on  a  sur 
face  us  =  const,  constitute  a  surface  S  which  meets  the  surfaces  us  =  const,  in 
asymptotic  lines  of  the  latter  and  geodesies  on  <S. 

6.  Show  that  the  asymptotic  lines  correspond  on  the  pseudospherical  surfaces 
of  a  triple  system  of  Bianchi. 

7.  Show  that  there  exist  triply  orthogonal  systems  for  which  the  surfaces  in  one 
family,  say  u$  —  const.,  are  spherical,  and  that  the  parameters  can  be  chosen  so  that 

HI  =  cosh  8,         Hz  =  sinh  6,         H3  =  US  —  . 

CUz 

Find  the  equations  of  Lame"  for  this  case. 

8.  Every  one-parameter  family  of  spheres  or  planes  is  a  family  of  Lame'. 

*Cf.  Bianchi,  Vol.  II,  p.  494. 


464     TRIPLY  ORTHOGONAL  SYSTEMS  OF  SURFACES 

9.  In  order  to  obtain  the  most  general  triply  orthogonal  system  for  which  the 
surfaces  in  one  family  are  planes,  one  need  construct  an  orthogonal  system  of 
curves  in  a  plane  and  allow  the  latter  to  roll  over  a  developable  surface,  in  which 
case  the  curves  generate  the  other  surfaces.  When  the  developable  is  given,  the 
determination  of  the  system  reduces  to  quadratures. 

10.  Show  that  the  most  general  triply  orthogonal  system  for  which  one  family 
of  Lame"  consists  of  spheres  passing  through  a  point  can  be  found  by  quadratures. 

11.  Show  that  a  family  of  parallel  surfaces  is  a  family  of  Lame'. 

12.  Show  that  the  triply  orthogonal  systems  for  which  the  curves  of  parameter 
«3  are  circles  passing  through  a  point  can  be  found  without  quadrature. 

13.  By  means  of  Ex.  6,  §  185,  show  that  for  a  system  of  Weingarten  of  constant 
curvature  the  principal  normals  to  the  curves  of  parameter  w3  at  the  points  of  meet 
ing  with  a  surface  u3  =  const,  form  a  normal  pseudospherical  congruence,  and  that 
the  surfaces  complementary  to  the  surfaces  w3  =  const,  and  their  orthogonal  tra 
jectories  constitute  a  system  of  Weingarten  of  constant  curvature. 

14.  By  means  of  Ex.  13  show  that  for  a  triple  system  arising  from  a  system  of 
Weingarten  of  constant  curvature  by  a  transformation  of  Combescure  the  osculat 
ing  planes  of  the  curves  w3  =  const.,  at  points  of  a  surface  us  =  const.,  envelop  a 
surface  S  of  the  same  kind  as  this  surface  M3  =  const.  ;  and  these  surfaces  S  and 
their  orthogonal  trajectories  constitute  a  system  of  the  same  kind  as  the  one  result 
ing  from  the  Combescure  transformation  of  the  given  system  of  Weingarten. 

15.  Show  that  a  necessary  condition  that  the  curves  of  parameter  u\  of  a  triple 
system  of  Bianchi  be  plane  is  that  w  satisfy  also  the  conditions 

d(*t  .  Set 

—  =  023  sm  w,        —  =  0i3  sin  w, 

di/2  dui 

where  023  and  0i8  are  independent  of  HI  and  w2  respectively  (cf.  Ex.  5,  p.  317). 
Show  that  if  0is  and  023  satisfy  the  conditions 


where  a  and  b  are  constants  and  U&  is  an  arbitrary  function  of  w8,  the  function  w, 
given  by  g023      a0i8 


COS  W  = 


determines  a  triply  orthogonal  system  of  Bianchi  of  the  kind  sought. 

16.  When  Z73  =  1  and  w  is  independent  of  u2,  the  first  and  fourth  of  equations 

(25)  may  be  replaced  by  gw 

—  =  sin  w. 
dui 

Show  that  for  a  value  of  «  satisfying  this  condition  and  the  other  equations  (25) 

the  expressions  /  rfadus          \       r^cosu, 

HI  =  cos  w  (  I  —  -  +  0i  )  -  /  —  :  -  du3  -f  0i, 
\J    sin  w  /     J     sin  w 


Hz  =  sin  w  (  C^J^  _(-  0A-  r03cZM8  +  02, 
\J    sin  w  /     J 

(  ffoduz          \  du 

HB~  (  I         -  +  0i    —  ' 

\  J   sin  w  /  5u3 


GENERAL  .EXAMPLES  465 

where  0i,  02,  03  are  functions  of  MI,  u2j  w3  respectively,  and  the  accent  indicates 
differentiation,  define  a  triply  orthogonal  system  for  which  the  surfaces  M3  =  const, 
are  molding  surfaces. 

17.  Under  what  conditions  do  the  functions 

d2« 


sin  w 


where  Z72  and  Z7s  are  functions  of  M2  and  M3  respectively,  determine  a  triply  orthog 
onal  system  arising  from  a  triple  system  of  Bianchi  by  a  transformation  of  Combes- 
cure  ?  Show  that  in  this  case  the  surfaces  w2  =  const,  are  spheres  of  radius  Z72,  and 
that  the  curves  of  parameter  M2  in  the  system  of  Bianchi  are  plane  or  spherical. 

18.  Prove  that  the  equations 


y  —  B(UI  —  b)mi(u2  —  b)m*(us  —  6)ms, 

Z  =  C(Ui  —  C)wli(lt2  —  C)w2(w3  —  c)'"3, 

where  A,  B,  C,  a,  6,  c,  mt-  are  constants,  define  space  referred  to  a  triple  system  of 
surfaces,  such  that  each  surface  is  cut  by  the  surfaces  of  the  other  two  families  in 
a  conjugate  system. 

19.  Given  a  surface  8  and  a  sphere  S;  the  circles  orthogonal  to  both  constitute 
a  cyclic  system  ;  hence  the  locus  of  a  point  upon  these  circles  which  is  in  constant 
cross-ratio  with  the  points  of  intersection  with  S  and  S  is  a  surface  Si  orthogonal 
to  the  circles  ;  Si  may  be  looked  upon  as  derived  from  S  by  a  contact  transformation 
which  preserves  lines  of  curvature  ;  such  a  transformation  preserves  planes  and 
spheres. 

20.  When  S  of  Ex.  19  is  a  cyclide  of  Dupin,  so  are  the  surfaces  Si,  and  also  the 
surface  which  is  the  locus  of  the  circles  which  meet  S  in  any  line  of  curvature  ; 
hence  all  of  these  surfaces  form  a  triple  system  of  cyclides  of  Dupin. 

21.  Given  three  functions  Z7,  defined  by 

Ui  =  imuf  +'2  mm  +  p^  (i  =  1,  2,  3) 

where  mt-,  Wj,  pi  are  constants  satisfying  the  conditions 

Smt-  =  0,         Snt-  =  0,         Spt-  =  0  ; 
and  given  also  the  function 

N=  tti(M2  -  u3)VUi  -f  a2  (M8  -  MI)  VU^  +  <*3  (MI  - 
+  pZniiUi  +  7  (PiMaWa  +  PZ^UI 
where  «,-,  0,  7  are  constants  ;  determine  under  what  condition  the  functions 

TT          W2  -  US  US  -  Ui  MI  -Ma 

&l—  -  ;=»  ««—  -  7='  ^33=  -  p= 

N^lfi  N^U2  N-VU3 

determine  a  triply  orthogonal  system.  Show  that  all  of  the  surfaces  are  isothermic, 
and  that  they  are  cyclides  of  Dupin. 

22.  Determine  whether  there  exist  triply  orthogonal  systems  of  minimal  surfaces. 


INDEX 


The  numbers  refer  to  pages.  References  to  an  author  and  his  contributions  are  made 
in  the  form  of  the  first  Bianchi  paragraph,  whereas  when  a  proper  name  is  part  of  a  title 
the  reference  is  given  the  form  as  in  the  second  Bianchi  paragraph. 


Acceleration,  15,  60 
Angle  between  curves,  74,  200 
Angle  of  geodesic  contingence,  212 
Applicable  surfaces,  definition,  100  ;  to 
the   plane,   101,    156 ;    invariance   of 
geodesic  curvature,  135 ;    invariance 
of  total  curvature,  156  ;   solution  of 
the  problem  of  determining  whether 
two   given    surfaces    are    applicable, 
321-326  ;    pairs  of,   derived    from  a 
given  pair,  349.    See  Deformation  of 
surfaces 

Area,  element  of,  75,  145 
Area  of  a  portion  of  a  surface,  145,  250 ; 

minimum,  222 

Associate  surfaces,  definition,  378  ;  de 
termination,  378-381;  of  a  ruled 
surface,  381;  of  the  sphere,  381;  ap 
plicable,  381;  of  the  right  helicoid, 
381 ;  of  an  isothermic  surface,  388  ; 
of  pseudospherical  surfaces,  390  ;  of 
.quadrics,  390,  391;  characteristic 
property,  425 

Asymptotic  directions,  definition,  \2S 
*Asymptotic  lines,  definition,  128  ;  para 
metric,  129, 189-194  ;  orthogonal,  129  ; 
straight,  140,  234  ;  spherical  represen 
tation,  144,  191-193;  preserved  by 
protective  transformation,  202  ;  pre 
served  in  a  deformation,  342-347 

Backhand,  transformation  of,  284-290 

Beltrami  (differential  parameters),  88, 
90;  (geodesic  curvature),  183;  (ruled 
W-surfaces),  299 ;  (applicable  ruled 
surfaces),  345  ;  (normal  congruences), 
403 

Bertrand  curves,  definition,  39  ;  proper 
ties,  39-41 ;  parametric  equations,  51 ; 
on  a  ruled  surface,  250  ;  deformation, 
348 

Bianchi  (theorem  of  permutability), 
286-288  ;  (surfaces  with  circular  lines 
of  curvature),  311;  (surfaces  with 


spherical  lines  of  curvature),  315 ; 
(associate  surfaces),  378  ;  (cyclic  con 
gruences  of  Ribaucour),  435  ;  (cyclic 
systems),  441 

Bianchi,  transformation  of,  280-283, 
290,  318,  320,  370,  456 ;  surfaces  of, 
370,  371,  442,  443,  445  ;  generalized 
transformation  of,  439  ;  triply  orthog 
onal  systems  of,  452-454,  464,  465 

Binormal  to  a  curve,  definition,  12 ; 
spherical  indicatrix,  50 

Bmormals  which  are  the  principal  nor 
mals  to  another  curve,  51 

Bonnet  (formula  of  geodesic  curvature), 
136 ;  (surfaces  of  constant  curvature), 
179;  (lines  of  curvature  of  Liouville 
type),  232  ;  (ruled  surf  aces),  248  ;  (sur 
faces  of  constant  mean  curvature), 
298 

IJour  (helicoids),  147;  (associate  isother 
mic  surfaces),  388 

Canal  surfaces,  definition,  68  ;  surfaces 
of  center,  186 

Catenoid,  definition,  150  ;  adjoint  sur 
face  of,  267 ;  surfaces  applicable  to,  318 

Cauchy,  problem  of,  265,  335 

Central  point,  243 

Central  plane,  244 

Cesaro  (moving  trihedral),  8S 

Characteristic  equation,  375 

Characteristic  function,,  374,  377 

Characteristic  lines,  13Q,  131 ;  paramet 
ric,  203 

Characteristics,  of  a  fanUly  of  surfaces, 
59-61 ;  of  the  tangent  pfones  to  a  sur--/ 
face,  126 

Christoffel  (associate  isothermic  sur 
faces),  388 

Christoffel  symbols,  definition,  152,  153  ; 
relations  between,  for  a  surface  and 
its  spherical  representation,  162,  193, 
201 

Circle,  of  curvature,  14  ;  osculating,  14 


*  References  to  asymptotic  lines,  geodesies,  lines  of  curvature,  etc.,  on  particular  kinds  of 
surfaces  are  listed  under  the  latter. 

467 


468 


INDEX 


Circles,  orthogonal  system  of,  in  the 
plane,  80,  97  ;  on  the  sphere,  301 

Circular  lines  of  curvature,  149,  310, 
316,  423,  446 

Circular  point  on  a  surface,  124 

Codazzi,  equations  of,  155-157,  161,  168, 
170,  189,  200 

Combescure  transformation,  of  curves, 
50  ;  of  triple  systems,  401-465 

Complementary  surface,  184,  185,  283, 
290,  370,  464 

Conforinal  representation,  of  two  sur 
faces,  98-100,  391;  of  a  surface  and 
its  spherical  representation,  143 ;  of 
a  surface  upon  itself,  101-103  ;  of  a 
plane  upon  itself,  104, 112 ;  of  a  sphere 
upon  the  plane,  109  ;  of  a  sphere  upon 
itself,  110,  111;  of  a  pseudospherical 
surface  upon  the  plane,  317 

Conformal-con jugate  representation  of 
two  surfaces,  224 

Congruence  of  curves,  426  ;  normal,  430 

Congruence  of  straight  lines  (rectilinear), 
definition,  392  ;  normal,  393,  398,  401, 
402,  403,  412,  422,  423,  437;  associate 
normal,  401-403,  411;  ruled  surfaces, 
393,  398,  401 ;  limit  points,  396  ;  prin 
cipal  surfaces,  396-398,  408 ;  principal 
planes,  396,  397;  developable*,  398, 
409,  414,  421,  432,  437;  focal  points, 
398,  399,  425;  middle  point,  399; 
middle  surface,  399,  401,  408,  413, 
421-424  ;  middle  envelope,  413,  415  ; 
focal  planes,  400,  401,  409,  416  ;  focal 
surfaces,  400,  406,  409-411,  412,  414, 
416,  420  ;  derived,  403-405,  411,  412  ; 
isotropic,  412,  413,  416;  of  Guichard, 
414,415,417,422,442 ;  pseudospherical, 
184,  415,  416,  464  ;  W-,  417-420,  422, 
424  ;  of  Ribaucour,  420-422,  424,  425, 
435,  442,  443  ;  mean  ruled  surfaces, 
422,  423,  425  ;  cyclic,  431-445 ;  spher 
ical  representation  of  cyclic,  432-433  ; 
cyclic  of  Ribaucour,  435,  442,  443  ; 
developables  of  cyclic,  437,  441 ; 
normal  cyclic,  437 

Conjugate  directions,  126,  173  ;  normal 
radii  in,  131 

Conjugate  system,  definition,  127,  223 ; 
parametric,  195,  203,  223,  224  ;  spher 
ical  representation  of,  200 ;  of  plane 
curves,  224  ;  preserved  by  projective 
transformation,  202  ;  preserved  in  a 
deformation,  338-342,  348,  349 

Conjugate  systems  in  correspondence,- 130 

Conoid,  right,  56,  58,  59,  68,  82,  98,  112, 
120,  195,  347 

Coordinates,  curvilinear,  on  a  surface, 
55  ;  curvilinear,  in  space,  447 ;  sym 
metric,  91-93  ;  tangential,  163,  194, 
201;  elliptic,  227 


Correspondence  with  orthogonality  of 
linear  elements,  374-377,  390 

Corresponding  conjugate  systems,  130 

Cosserat  (infinitesimal  deformation), 
380,  385 

Cross-ratio,  of  four  solutions  of  a  Riccati 
equation,  26  ;  of  points  of  intersection 
of  four-curved  asymptotic  lines  on  a 
ruled  surface,  249 ;  of  the  points  in 
which  four  surfaces  orthogonal  to  a 
cyclic  system  meet  the  circles,  429 

Cubic,  twisted,  4,  8,  11,  12,  15,  269 

Curvature,  first,  of  a  curve,  9;  radius 
of,  9;  center  of,  14;  circle  of,  14; 
constant,  22,  38,  51 

Curvature,  Gaussian,  123 ;  geodesic  (see 
Geodesic) 

Curvature,  mean,  of  a  surface,  123,  126, 
145  ;  surfaces  of  constant  (see  Sur 
face) 

Curvature,  normal,  of  a  surface,  radius 
of,  118,  120,  130,  131,  150;  principal 
radii  of,  119,  120,  291,  450  ;  center  of, 
118,  150;  principal  centers  of,  122 

Curvature,  second,  of  a  curve,  16  ;  con 
stant,  50.  See  Torsion 

Curvature,  total,  of  a  surface,  123,  126, 
145,  155,  156,  160,  172,  186,  194,  208, 
211  ;  radius  of,  189  ;  surfaces  of  con 
stant  (see  Surface) 

Curve,  definition,  2;  of  constant  first 
curvature,  22,  38,  51;  of  constant 
torsion,  50  ;  form  of  a,  18 

Cyclic  congruences.    See  Congruences 

Cyclic  system,  426-445  ;  definition,  426  ; 
of  equal  circles,  430,  443  ;  surfaces 
orthogonal  to,  436,  437,  444,  457; 
planes  envelop  a  curve,  439,  440 ; 
planes  through  a  point,  440,  441 ; 
planes  depend  on  one  parameter,  442  ; 
triple  system  associated  with  a,  446  ; 
associated  with  a,  triple  system,  457; 
458 

Cyclides  of  Dupin,  188,  312-314,  412, 
422,  465 

D,  Z7,  7)",  definition,  115  ;  for  the  mov 
ing  trihedral,  174 

A  Jb'i  &"•>  definition,  386 

Darboux  (moving  trihedral),  168,  169, 
170 ;  (asymptotic  lines  parametric), 
191 ;  (conjugate  lines  parametric), 
195  ;  (lines  of  curvature  preserved  by 
an  inversion),  196  ;  (asymptotic  lines 
and  conjugate  systems  preserved  by 
projective  transformation),  202  ;  (geo 
desic  parallels),  216,  217  ;  (genera 
tion  of  new  surfaces  of  Weingarten), 
298 ;  (generation  of  surfaces  with 
plane  lines  of  curvature  in  both  sys 
tems),  304 ;  (general  problem  of 


INDEX 


469 


deformation),  332 ;  (surfaces  appli 
cable  to  paraboloids),  367 ;  (triply 
orthogonal  systems),  458-461 

Darboux,  twelve  surfaces  of,  391;  de 
rived  congruences  of,  404,  405 

Deformation  of  surfaces  (see  Applicable 
surfaces) ;  of  surfaces  of  revolution 
(see  Surfaces  of  revolution) ;  of  mini 
mal  surfaces,  264,  269,  327-330  ;  of 
surfaces  of  constant  curvature,  321- 
323  ;  general  problem,  331-333  ;  which 
changes  a  curve  on  the  surface  into  a 
given  curve  in  space,  333-336  ;  which 
preserves  asymptotic  lines,  336,  342, 
343  ;  which  preserves  lines  of  curva 
ture,  336-338,  341 ;  which  preserves 
conjugate  systems,  338-342,  349,  350, 
443 ;  of  ruled  surfaces,  343-348,  350, 
367;  method  of  Weingarten,  353-369 ; 
of  paraboloids,  348,  368,  369  ;  of  the 
envelope  of  the  planes  of  a  cyclic 
system,  429,  430 

Developable  surface,  definition,  61 ; 
equation,  64  ;  particular  kinds,  69 ; 
rectifying,  62,  64,  112,  209  ;  polar,  64, 
65,  112,  209  ;  applicable  to  the  plane, 
101,  156,  219,  321,  322 ;  formed  by  nor 
mals  to  a  surface  at  points  of  a  line  of 
curvature,  122  ;  principal  radii,  149 ; 
total  curvature,  156,  250  ;  geodesies 
on  a,  224,  268,  318,  322  ;  fundamental 
property,  244;  of  a  congruence  (see 
Congruence) 

Dextrorsum,  19 

Differential  parameters,  of  the  first  order, 
84-88,  90,  91,  120,  160,  166,  186  ;  of 
the  second  order,  88-91,  160,  165,  166, 
186 

Diui  (spherical  representation  of  asymp 
totic  lines),  192;  (surf  aces  of  Liouville), 
214  ;  (ruled  TF-surfaces),  299 

Dini,  surface  of,  291,  318 

Director-cone  of  a  ruled  surface,  141 

Director-developable  of  a  surface  of 
Monge,  305 

Directrix  of  a  ruled  surface,  241 

Dobriner  (surfaces  with  spherical  lines 
of  curvature),  315 

Dupin  (triply  orthogonal  systems),  449 

Dupin,  indicatrix  of,  124-126,  129,  150  ; 
cyclide  of  (see  Cyclide) ;  theorem  of 
Malus  and,  403 

jE,  F,  G,  definition,  70  ;  for  the  moving 

trihedral,  174 
&•>  &•>  ^  definition,  141 ;  for  the  moving 

trihedral,  174 
e,/,/,  flr,  definition,  393 
Edge  of  regression,  43,  60,  69 


Element,  of  are.a,  75,  145 ;  linear  (see 
Linear  element) 

Ellipsoid,  equations,  228  ;  normal  sec 
tion,  234 ;  polar  geodesic  system, 
236-238;  umbilical  geodesies,  236, 
267;  surface  corresponding  with  par 
allelism  of  tangent  plane,  269.  See 
Quadrics 

Elliptic  coordinates,  227 

Elliptic  point  of  a  surface,  125,  200 

Elliptic  type,  of  pseudospherical  sur 
faces,  274 ;  of  surfaces  of  Bianchi, 
370,  371 

Enneper  (torsion  of  asymptotic  lines), 
140 ;  (equations  of  a  minimal  surface) , 
256 

Enneper,  minimal  surface  of,  269;  sur 
faces  of  constant  curvature  of ,  317,  320 

Envelope,  definition,  59,  60 ;  of  a  one- 
parameter  family  of  planes,  61-63, 
64,  69,  442  ;  of  a  one-parameter  fam 
ily  of  spheres,  66-69  ;  of  a  two-param 
eter  family  of  planes,  162,  224,  426, 
439;  of  geodesies,  221;  of  a  two- 
parameter  family  of  spheres,  391,  444 

*  Equations,  parametric,  1,  2,  52,  53; 
of  a  curve,  1,  2,  3,  21;  of  a  surface, 
52,  53,  54 

Equidistance,  curves  of,  456 

Equidistantial  system,  187,  203 

Equivalent  representation  of  two  sur 
faces,  113,  188 

Euler,  equation  of,  124,  221 

Evolute,  of  a  curve,  43,  45-47 ;  of  a 
surface,  180,  415  (see  Surface  of 
center) ;  of  the  quadrics,  234 ;  mean, 
of  a  surface,  165,  166,  372 

F.    SeeE 

&    See£ 

//'.    Seee 

Family,  one-parameter,  of  surfaces,  59, 
446,  447,  451,  452,  457-461;  of  planes, 
61-64,  69,  442,  463  ;  of  spheres,  66-69, 
309,  319,  463 ;  of  curves,  78-80 ;  of  geo 
desies,  216,  221 

Family,  two-parameter,  of  planes,  162, 
224,  426,  439 ;  of  spheres,  391,  444 

Family  of  Lame",  461,  463,  464 

Focal  conic,  226,  234,  313,  314 

Focal  planes,  400,  401,  409,  416 

Focal  points,  398,  399,  425 

Focal  surface,  of  a  congruence,  400  ; 
reduces  to  a  curve,  406,  412  ;  funda 
mental  quantities,  409-411 ;  develop 
able,  412;  met  by  developables  in 
lines  of  curvature,  414 ;  of  a  pseudo- 
spherical  congruence,  416;  infinitesi 
mal  deformation  of,  420 ;  intersect,  423 


*  For  references  such  as  Equations  of  Codazzi,  see  Codazzi. 


4TO 


INDEX 


Form  of  a  curve,  18 

Frenet-Serret  formulas,  17 

Fundamental  equations  of  a  congruence, 
406,  407 

Fundamental  quadratic  form,  of  a  sur 
face,  first,  7 1  ;  of  a  surface,  second, 
115;  of  a  congruence,  393 

Fundamental  quantities,  of  the  first 
order,  71;  of  the  second  order,  115 

Fundamental  theorem,  of  the  theory  of 
curves,  24 ;  of  the  theory  of  surfaces, 
159 

G.    See  E 

&    8ee£ 

g.    See  e 

Gauss  (parametric  form  of  equations), 
60  ;  (spherical  representation),  141  ; 
(total  curvature  of  a  surface),  155  ; 
(geodesic  parallels),  200;  (geodesic  cir 
cles)  207  ;  (area  of  geodesic  triangle) 
209 

Gauss,  equations  of,  154,  155,  187 

Generators,  of  a  developable  surface, 
41 ;  of  a  surface  of  translation,  198  ; 
of  a  ruled  surface,  241 

Geodesic  circles,  207 

Geodesic  contingence,  angle  of,  212 

Geodesic  curvature,  132,  134,  135,  13(5, 
140,  213,  223  ;  radius  of,  132,  150,  151, 
174,  170,  209,  411 ;  center  of,  132,  225, 
423 ;  invariance  of,  135 ;  curves  of 
constant,  137,  140,  187,  223,  319 

Geodesic  ellipses  and  hyperbolas,  213- 
215,  225 

Geodesic  parallels,  207 

Geodesic  parameters,  207 

Geodesic  polar  coordinates,  207-209, 
230,  276 

Geodesic  representation,  225,  317 

Geodesic  torsion,  137-140,  174,  176 ; 
radius  of,  138,  174,  176 

Geodesic  triangle,  209,  210 

*  Geodesies,  definition,  133  ;  plane,  140  ; 
equations  of,  204,  205,  215-219;  on 
surfaces  of  negative  curvature,  211  ; 
on  surfaces  of  Liouville,  218,  219 

Goursat,  surfaces  of,  306,  372 

Guichard  (spherical  representation  of 
the  developables  of  a  congruence), 
409;  (congruences  of  Ribaucour),  421 

Guichard,  congruences  of,  414,  415,  417, 
422,  442 

I/,  definition,  71 
//,  definition,  142 
Hamilton,  equation  of,  397 
Hazzidakis,  transformation  of,  278,  279, 
338 


Helicoid,  general,  146-148  ;  parameter 
of,  140  ;  meridian  of,  140  ;  geodesies, 
149,  151,  209 ;  surfaces  of  center  of, 
186  ;  pseudospherical,  291 ;  is  a  IP-sur 
face,  300  ;  minimal,  329,  331  ;  appli 
cable  to  a  hyperboloid,  347 

Helicoid,  right,  146,  148,  203,  247,  250, 
260,  267,  330,  347,  381,  422 

Helix,  circular,  2,  41,  45,  203  ;  cylindri 
cal,  20,  21,  29,  30,  47,  64 

Henneberg,  surface  of,  267 

Hyperbolic  point,  125,  200 

Hyperbolic  type,  of  pseudospherical  sur 
face,  273  ;  of  surface  of  Bianchi,371, 379 

Hyperboloid,  equations,  228 ;  fundamen 
tal  quantities,  228-230;  evolute  of, 
234  ;  of  revolution,  247,  348 ;  lines  of 
striction,  268  ;  deformation  of,  347, 
348.  See  Quadrics 

Indicatrix,  of  Dupin  (seel)upin);  spheri 
cal  (.see  Spherical) 

Infinitesimal  deformation  of  a  surface, 
373,  385-387 ;  generatrices,  373,  420  ; 
of  a  right  helicoid,  381  ;  of  ruled  sur 
faces,  381  ;  in  which  lines  of  curva 
ture  are  preserved,  387,  391;  of  the 
focal  surfaces  of  a  TF-congruence,  420 

Intrinsic  equations  of  a  curve,  23,  29, 
30,  30 

Invariants,  differential,  85-90  ;  of  a  dif 
ferential  equation,  380,  385,  406 

Inversion,  definition,  190 ;  preserves 
lines  of  curvature,  190  ;  preserves  an 
isotherm ic  system  of  lines  of  curva 
ture,  391  ;  preserves  a  triply  orthog 
onal  system,  403.  See  Transformation 
by  reciprocal  radii 

Involute,  of  a  curve,  43-45,  311 ;  of  a 
surface,  180,  184,  300 

Isometric  parameters.  See  Isothermic 
parameters  ,-' 

Isometric  representation,  100,  113 

Isothermal-conjugate  systems  of  curves, 
198-200;  spherical  representation,  202 ; 
formed  of  lines  of  curvature,  147,  203, 
233,  278  ;  on  associate  surfaces,  300 

Isothermal -orthogonal  system.  See  Iso 
thermic  orthogonal  system 

Isothermic  orthogonal  systems,  93-98, 
209,  252,  254 ;  formed  of  lines  of  curva 
ture  (see  Isothermic  surface) 

Isothermic  parameters,  93-97,  102 

Isothermic  surface,  108,  159,  232,  253, 
269,  297,  387-389,  391,  425,  465 

Isotropic  congruence,  412,  413,  416,  422- 
424 

Isotropic  developable,  72,  171,  412,  424 

Isotropic  plane,  49 


*  See  footnote,  p.  467. 


INDEX 


471 


Jacob!  (geodesic  lines),  217 
Joachimsthal    (geodesies   and    lines   of 

curvature  on  central  quadrics),  240 
Joachimsthal,  theorem  of,  140  ;  surfaces 

of,  308,  309,  319 

Kummer  (rectilinear  congruences),  392 

Lagrange  (minimal  surfaces),  251 

Lame'  (differential  parameters),  85 

Lame",  equations  of,  449  ;  family  of,  401, 
463,  464 

Lelieuvre,  formulas  of,  193,  195,  417, 
419,  420,  422 

Lie  (surfaces  of  translation),  197,  198  ; 
(double  minimal  surfaces),  259  ;  (lines 
of  curvature  of  JF-surfaces),  293 

Lie,  transformation  of,  289,  297 

Limit  point,  396,  399 

Limit  surface,  389 

Line,  singular,  71 

*Line  of  curvature,  definition,  121,  122, 
128;  equation  of,  121,  171,  247;  par 
ametric,  122,  151,  186  ;  normal  cur 
vature  of,  121, 131 ;  geodesic  torsion  of, 
139 ;  geodesic,  140 ;  two  surfaces  inter 
secting  in,  140 ;  spherical  representa 
tion  of,  143, 148, 150;  osculating  plane, 
148;  plane,  149,  150,  201,  305-314, 
319,  320,  463 ;  plane  in  both  systems, 
269,  300-304,  319,  320  ;  spherical,  149, 
314-317,  319,  320,  465  ;  circular,  149, 
310-314,  316,  446;  on  an  isothermic 
surface,  389, 

Line  of  striction,  243,  244,  248,  268,  348, 
351,  352,  369,  401,  422 

Linear  element,  of  a  curve,  4,  5  ;  of  a 
surface,  42,  71,  171;  of  the  spherical 
representation,  141, 173,  393;  reduced 
form,  353  ;  of  space,  447 

Lines  of  length  zero.    See  Minimal  lines 

Lines  of  shortest  length,  212,  220 

Liouville  (form  of  Gauss  equation),  187 ; 
(angle  of  geodesic  contingence),  212 

Liouville,  surfaces  of,  214,  215,  218,  232 

Loxodromic  curve,  78, 108,  112, 120, 131, 
140,  209 

Mainardi,  equations  of,  156 

Malus  and  Dupin,  theorem  of,  403 

v.  Mangoldt  (geodesies  on  surfaces  of 

positive  curvature),  212 
Mean  curvature,  123,  126,  145 
Mean  evolute,  165,  166,  372 
Mean  ruled  surfaces  of  a  congruence, 

422,  423,  425 
Mercator  chart,  109 
Meridian,   of  a  surface  of   revolution, 

107;  of  a  helicoid,  146 


Meridian  curve  on  a  surface,  260 

Meusnier,  theorem  of,  118 

Middle  envelope  of  a  congruence,  413, 
415 

Middle  point  of  a  line  of  a  congruence, 
399 

Middle  surface  of  a  congruence,  399, 
401,  408,  413,  421-424 

Minding  (geodesic  curvature),  222,  223 

Minding,  problem  of,  321,  323,  326  ; 
method  of,  344 

Minimal  curves,  6,  47,  49,  255,  257 ; 
on  a  surface,  81,  82,  85,  91,  254-265, 
318,  391 ;  on  a  sphere,  81,  257,  364-366 
390 

Minimal  straight  lines,  48,  49,  260 

Minimal  surface,  definition,  129,  251; 
asymptotic  lines,  129,  186,  195,  254, 
257,  269 ;  spherical  representation, 
143,  251-254;  ruled,  148;  helicoidal, 
149,  330,  331  ;  of  revolution,  160 ; 
parallel  plane  sections  of,  160 ;  mini 
mal  lines,  177,  186,  254-265;  lines  of 
curvature,  186,  253,  257,  264,  269 ; 
double,  258-260  ;  algebraic,  260-262  ; 
evolute,  260,  372  ;  adjoint,  254,  263, 
267,  377;  associate,  263,  267,  269,  330, 
381;  of  Scherk,  260;  of  Henneberg, 
267;  of  Enneper,  269;  deformation 
of,  264,  327-329,  349,  381 ;  determi 
nation  of,  265,  266  ;  geodesies,  267 

Molding  surface,  definition,  302 ;  equa 
tions  of,  307,  308  ;  lines  of  curvature, 
307,  308,  320  ;  applicable,  319,  338  ; 
associate  to  right  helicoid,  381 ;  nor 
mal  to  a  congruence  of  Ribaucour, 
422 

Molding  surfaces,  a  family  of  Lame"  of, 
465 

Monge  (equations  of  a  surface),  64 ; 
(molding  surfaces),  302 

Monge,  surfaces  of,  305-308,  319 

Moving  trihedral  for  a  curve,  30-33  ; 
applications  of,  33-36,  39,  40,  64-68 

Moving  trihedral  for  a  surface,  166-170  ; 
rotationsof,  169;  applications  of,  171- 
183, 281-288, 336-338, 352-364, 426-442 

Normal,  principal,  definition,  12;  par 
allel  to  a  plane,  16,  21 

Normal  congruence  of  lines  (see  Con 
gruence)  ;  of  curves  (see  Congruence) 

Normal  curvature  of  a  surface.  See 
Curvature 

Normal  plane  to  a  curve,  8,  15,  65 

Normal  section  of  a  surface,  118,  234 

Normal  to  a  curve,  12 

Normal  to  a  surface,  57,  114,  117,  120, 
121,  141,  195 


*  See  footnote,  p.  467. 


472 


INDEX 


Normals,  principal,  which  are  principal 
normals  of  another  curve,  41 ;  which 
are  binormals  of  another  curve,  51 

Order  of  contact,  8,  21 

Orthogonal  system  of  curves,  75,  77, 
80-82,  91,  119,  129,  177,  187;  par 
ametric,  75,  93,  122,  134  ;  geodesies, 
187  ;  isothermic  (see  Iso thermic) 

Orthogonal  trajectories,  of  a  one-param 
eter  family  of  planes,  35,  451  ;  of  a 
family  of  curves,  50,  79,  95,  112,  147, 
149,  150  ;  of  a  family  of  geodesies, 
216  ;  of  a  family  of  surfaces,  446,  451, 
452,  456,  457,  460,  463,  464 

Osculating  circle,  14,  21,  65 

Osculating  plane,  definition,  10 ;  equa 
tion  of,  11  ;  stationary,  18  ;  meets  the 
curve,  19 ;  passes  through  a  fixed  point, 
22  ;  orthogonal  trajectories  of,  35  ;  of 
edge  of  regression,  57  ;  of  an  asymp 
totic  line,  128  ;  of  a  geodesic,  133 

Osculating  planes  of  two  curves  parallel, 
50 

Osculating  sphere,  37,  38,  47,  51,  65 

Parabolic  point  on  a  surface,  125 

Parabolic  type,  of  pseudospherical  sur 
faces,  274  ;  of  surfaces  of  Bianchi, 
370,  371,  442,  443,  445 

Paraboloid,  a  right  conoid,  56 ;  tangent 
plane,  112  ;  asymptotic  lines,  191,  233; 
a  surface  of  translation,  203  ;  equa 
tions,  230,  330  ;  fundamental  quanti 
ties,  231 ;  lines  of  curvature,  232,  240  ; 
evolute  of,  234  ;  of  normals  to  a  ruled 
surface,  247 ;  line  of  striction,  268  ; 
deformation  of,  348,  349,  367-369, 
372 ;  congruence  of  tangents,  401. 
See  Quadrics 

Parallel,  geodesic,  86,  207  ;  on  a  surface 
of  revolution,  107 

Parallel  curves,  44 

Parallel  surface,  definition,  177  ;  lines 
of  curvature,  178  ;  fundamental  quan 
tities,  178;  of  surface  of  constant  cur 
vature,  179  ;  of  surface  of  revolution, 
185 

Parallel  surfaces,  a  family  of  Lame"  of, 
446 

Parameter,  definition,  1 ;  of  distribution, 
245,  247,  268,  348,  424,  425 

Parametric  curves,  54,  55 

Parametric  equations.    See  Equations 

Plane  curve,  condition  for,  2,  16  ;  curv 
ature,  15  ;  equations,  28,  49 ;  intrinsic 
equations,  36 

Plane  curves  forming  a  conjugate  sys 
tem,.  224 

Plane  lines  of  curvature.  See  Lines  of 
curvature 


Point  of  a  surface,  singular,  71 ;  elliptic, 
125,  200 ;  hyperbolic,  125,  200  ;  para 
bolic,  125;  focal  (see  Focal) ;  middle 
(see  Middle);  limit  (see  Limit) 
Polar  developable,  64,  65,  112,  209 
Polar  line  of  a  curve,  15,  38,  46 
Principal  directions  at  a  point,  121 
Principal  normal  to  a  curve.  See  Normal 
Principal  planes  of  a  congruence,  396, 397 
Principal  radii  of  normal  curvature,  lit), 

120,  291,  450 
Principal  surfaces  of  a  congruence,  390- 

398,  408 

Projective  transformation,  preserves  os 
culating  planes,  49  ;  preserves  asymp 
totic  lines  and  conjugate  systems,  202 
Pseudosphere,  274,  290 
Pseudospherical   congruence,  415,  416, 

464  ;  normal,  184 

Pseudospherical  surface,  definition,  270  ; 
asymptotic  lines,  190,  290,  414  ;  lines 
of  curvature,  190,  203,  280,  320 ;  geo 
desies,  275-277,  283,  317,  318  ;  defor 
mation,  277,  323  ;  transformations  of, 
280-290,  318,  320,  370,  45(5 ;  of  Dini, 
291,  318;  of  Enneper,  317,  820;  evo 
lute,  318  ;  involute,  318  ;  surfaces  with 
the  same  spherical  representation  of 
their  lines  of  curvature  as,  320,  371, 
437,  439,  443,  444.  See  Surface  of 
constant  total  curvature 
Pseudospherical  surface  of  revolution, 
of  hyperbolic  type,  273  ;  of  elliptic 
type,  274  ;  of  parabolic  type,  274 
Pseudospherical  surfaces,  a  family  of 
Lam6  of,  452-456,  464 

Quadratic  form.  See  Fundamental 
Quadrics,  confocal,  226,401 ;  fundamen 
tal  quantities,  229  ;  lines  of  curvature, 
233,  239,  240  ;  asymptotic  lines,  233  ; 
geodesies,  234-236,  239,  240 ;  associate 
surfaces,  390,  391  ;  normals  to,  422. 
See  Ellipsoid,  Hyperboloid,  Paraboloid 

Representation,  conformal  (see  Con- 
formal);  isometric,  100,  113;  equiv 
alent,  113,  188;  Gaussian,  141; 
conformal-con  jugate,  224  ;  geodesic, 
225,  317  ;  spherical  (see  Spherical) 
Revolution,  surfaces  of.  See  Surface 
Ribaucour  (asymptotic  lines  on  surfaces 
of  center),  184 ;  (cyclic  systems  of 
equal  circles),  280;  (limit  surfaces), 
389 ;  (middle  envelope  of  an  isotropic 
congruence),  413 ;  (cyclic  systems), 
426,  428,  432  ;  (deformation  of  the 
envelope  of  the  planes  of  a  cyclic 
system),  429,  430 ;  (cyclic  systems 
associated  with  a  triply  orthogonal 
system),  457 


INDEX 


473 


Ribaucour,  congruence  of,  420-422,  424, 
425,  435,  442,  443  ;  triple  systems  of, 
452,  455,  463 

Riccati  equation,  25,  26,  50,  248,  429 

Rodrigues,  equations  of,  122 

*  Ruled  surface,  definition,  241 ;  of  tan 
gents  to  a  surface,  188 ;  generators, 
241;  directrix,  241;  linear  element, 
241,  247  ;  director-cone,  241 ;  line  of 
striction,  243,  244,  248,  268,  348,  351, 
352,  309,  401,  422  ;  central  point,  243  ; 
central  plane,  244 ;  parameter  of  dis 
tribution,  245,  247,  208,  348,  424,  425  ; 
doubly,  234;  normals  to,  195,  247; 
tangent  plane,  246,  247,  268  ;  total 
curvature,  247 ;  asymptotic  lines,  248- 
250  ;  mean  curvature,  249 ;  lines  of 
curvature,  250,  268  ;  conjugate,  268 ; 
deformation,  343-348,  350,  367;  spher 
ical  indicatrix  of,  351;  infinitesimal 
deformation,  381 ;  of  a  congruence, 
393-395,  398,  401,  422,  423.  See  Right 
conoid,  Hyperboloid,  Paraboloid 

Scheffers  (equations  of  a  curve),  28 

Scherk,  surface  of,  260 

Schwarz,  formulas  of,  264-267,  269 

Singular  line  of  a  surface,  71 

Singular  point  of  a  surface,  71 

Sinistrorsum,  19 

Sphere,  equations,  62,  77,  81 ;  minimal 
lines,  81 ;  conformal  representation, 
109-111 ;  equivalent  representation, 
113 ;  fundamental  quantities,  116, 
171;  principal  radii,  120;  asymptotic 
lines,  223,  422 

Spheres,  family  of.    See  Family 

Spherical  curve,  36,  38,  47,  50,  149, 
314-316,  317,  319,  320,  465 

Spherical  indicatrix,  of  the  tangents  to 
a  curve,  9, 13,  50, 177 ;  of  the  binormals 
to  a  curve,  50,  177 ;  of  a  ruled  surface, 
351 

Spherical  representation  of  a  congruence, 
definition,  393 ;  principal  surfaces,  397, 
408  ;  developables,  409,  412-414,  422, 
432-435,  437,  441 

Spherical  representation  of  a  surface, 
definition,  141 ;  fundamental  quan 
tities,  141-143,  160-165,  173;  lines 
of  curvature,  143,  148,  150,  151,  188, 

201,  253,  279,  280,  292,  296,  301,  302, 
308,  314,  315,  320,  371,  387,  437,  442- 
445  ;  asymptotic  lines,  144,  148,  191- 
195,  254,  340,  390,  414  ;  area  of  closed 
portion,  145 ;  conjugate  system,  200- 

202,  257,  385 


Spherical  representation  of  an  axis  of 
a  moving  trihedral,  354 

Spherical  surface,  definition,  270;  par 
allels  to,  179 ;  of  revolution,  270-272  ; 
geodesies,  275-279,  318 ;  deformation, 
276,  323 ;  lines  of  curvature,  278  ; 
transformation,  278-280,  297 ;  invo 
lute,  300  ;  of  Enneper,  317  ;  surface 
with  the  same  spherical  representation 
of  its  lines  of  curvature  as,  338.  See 
Surface  of  constant  total  curvature 

Spherical  surfaces,  a  family  of  Lame"  of, 
463 

Spiral  surface,  definition,  151 ;  gener 
ation,  151  ;  lines  of  curvature,  151 ; 
minimal  lines,  151 ;  asymptotic  lines, 
151 ;  geodesies,  219  ;  deformation,  349 

Stereographic  projection,  110,  112 

Superosculating  circle,  21 

Superosculating  lines  on  a  surface. 
187 

t  Surface,  definition,  53 

Surface,  limit,  389 

\  Surface  of  center,  definition,  179 ;  met 
by  developables  in  a  conjugate  sys 
tem,  180, 181 ;  fundamental  quantities, 
181, 182  ;  total  curvature,  183  ;  asymp 
totic  lines,  183,  184  ;  lines  of  curva 
ture,  183,  184  ;  a  curve,  186,  188,  308- 
314 ;  developable,  186,  305-308 

Surface  of  constant  mean  curvature, 
definition,  179 ;  parallels  to,  179  ;  lines 
of  curvature,  296-298 ;  transforma 
tion,  297  ;  deformation,  298  ;  minimal 
curves,  318 

Surface  of  constant  total  curvature, 
definition,  179 ;  area  of  geodesic  tri 
angle,  219  ;  geodesies,  224 ;  lines  of 
curvature,  317;  asymptotic  lines,  317; 
spherical  representation,  372.  See 
Pseudospherical  surface  and  Spheri 
cal  surface 

Surface  of  reference,  392 

Surface  of  revolution,  definition,  107  ; 
fundamental  quantities,  107,  147 ; 
loxodromic  curve  (see  Loxodromic) ; 
deformation,  108,  112,  147,  149,  260, 
276,  277,  283,  326-331,  341,  349-350, 
362-364,  369,  370,  372,  444;  partic 
ular,  111,  160,  320  ;  equivalent  repre 
sentation,  113 ;  lines  of  curvature,  126 ; 
asymptotic  lines,  131 ;  parallel  sur 
faces,  185 ;  geodesies,  20.5,  209,  224 

Surface  of  translation,  definition,  197, 
198;  equations,  197;  asymptotic  lines, 
198;  generators,  198, 203;  deformation, 
349,  350  ;  associate  surface,  381,  390; 


*  Tim  reference  is  to  nondevelopable  ruled  surfaces.    For  developable  ruled  surfaces,  see 
Developables. 

t  For  references  such  as  Surface  of  Bianchi,  see  Bianchi. 

I  Surfaces  of  center  of  certain  surfaces  are  referred  to  under  these  surfaces. 


474 


INDEX 


congruence  of  tangents,  406  ;  middle 
surface  of  a  >F-congruence,  422,  424 

Surface  with  plane  lines  of  curvature. 
See  Lines  of  curvature 

Surface  with  spherical  lines  of  curvature. 
See  Lines  of  curvature 

Surface  with  the  same  spherical  repre 
sentation  of  its  lines  of  curvature  as  a 
pseudospherical  surface.  See  Pseudo- 
spherical  surface 

Surface  with  the  same  spherical  repre 
sentation  of  its  lines  of  curvature  as 
a  spherical  surface.  See  Spherical  sur 
face 

Surfaces  of  revolution,  a  family  of  Lame" 
of,  451 

Tangent  plane  to  a  surface,  definition, 
50, 114;  equation,  57;  developable  sur 
face,  67;  distance  to,  114  ;  meets  the 
surface,  123  ;  characteristic  of,  126 ; 
is  the  osculating  plane  of  asymptotic 
line,  128 

Tangent  surface  of  a  curve,  41-44,  57; 
applicable  to  the  plane,  101,  150 

Tangent  to  a  curve,  6,  7,  41),  50,  51) ; 
spherical  indicatrix  of,  9,  13,  50,  177 

Tangent  to  a  surface,  112 

Tangential  coordinates,  103,  104,  201 

Tetrahedral  surface,  definition,  207 ; 
asymptotic  lines,  207 ;  deformation, 
341 

Tetrahedral  surfaces,  triple  system  of, 
465 

Tore,  124 

Torsion,  geodesic,  137-140,  174,  170 

Torsion  of  a  curve,  definition,  10  ;  radius 
of,  16,  17,  21;  of  a  plane  curve,  10  ; 
sign  of,  19;  constant,  60;  of  asymp 
totic  line,  140 

Tractrix,  equations,  35  ;  surface  of  revo 
lution  of,  274,  290  ;  helicoid  whose 
meridian  is  a,  291 


*  Transformation,  of  curvilinear  coordi 
nates,  53-55,  73,  74  ;  of  rectangular 
coordinates,  72  ;  by  reciprocal  radii, 
104,  196,  203  (see  Inversion) ;  project- 
ive  (see  Projective) 

Triply  orthogonal  system  of  surfaces, 
definition,  447;  associated  with  a  cyc 
lic  system,  440  ;  fundamental  quan 
tities,  447-451  ;  with  one  family  of 
surfaces  of  revolution,  451,  452  ;  of 
Kibaucour,  452,  403  ;  of  Bianchi,  452- 
454,  404,  465  ;  of  Weingarten,  455, 
456,  403,  404  ;  transformation  of,  462, 
463  ;  with  one  family  of  molding  sur 
faces,  405  ;  of  cyclides  of  Dupin,  405  ; 
of  isothermic  surfaces,  405 

Umbilical  point  of  a  surface,  definition, 
120  ;  of  quadrics,  230,  232,  234,  230- 
238,  240,  207 

Variation  of  a  function,  82,  83 

Voss,  surface  of,  341,  390,  415,  442,  443 

W-congruence,  417-420,  422,  424 

>F-surface,  definition,  291 ;  fundamental 
quantities,  291-293  ;  particular,  291, 
300,  318, 319;  spherical  representation, 
292;  lines  of  curvature,  293;  evolute, 
294,  295,  318,  319;  of  Weingarten, 
298,  424;  ruled,  299,  319 

Weierstrass  (equations  of  a  minimal  sur 
face),  200  ;  (algebraic  minimal  sur 
faces),  201 

Weingarten  (tangential  coordinates). 
103  ;  (geodesic  ellipses  and  hyperbo 
las),  214;  (>F-surfaces),  291,  292,  294  ; 
(infinitesimal  deformation),  374,  387  ; 
(lines  of  curvature  on  an  isothermic 
surface),  389 

Weingarten,  surface  of,  298, 424 ;  method 
of,  353-372  ;  triple  system  of,  455,  450, 
403,  464 


*  For  references  such  as  Transformation  of  BJickluud,  see  Bitcklund. 


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